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Theorem alephfplem4 4879
Description: Lemma for alephfp 4880.
Hypothesis
Ref Expression
alephfplem.1 |- H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)
Assertion
Ref Expression
alephfplem4 |- U.(H"om) e. ran aleph
Distinct variable group:   x,y
Proof of Theorem alephfplem4
StepHypRef Expression
1 ffnfv 3819 . . . 4 |- (H:om-->ran aleph <-> (H Fn om /\ A.z e. om (H` z) e. ran aleph))
2 frfnom 3942 . . . . 5 |- (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) Fn om
3 alephfplem.1 . . . . . 6 |- H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om)
4 fneq1 3574 . . . . . 6 |- (H = (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) -> (H Fn om <-> (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) Fn om))
53, 4ax-mp 7 . . . . 5 |- (H Fn om <-> (rec({<.x, y>. | y = (aleph` x)}, (aleph` (/))) |` om) Fn om)
62, 5mpbir 190 . . . 4 |- H Fn om
73alephfplem3 4878 . . . . 5 |- (z e. om -> (H` z) e. ran aleph)
87rgen 1695 . . . 4 |- A.z e. om (H` z) e. ran aleph
91, 6, 8mpbir2an 729 . . 3 |- H:om-->ran aleph
10 ssun2 2190 . . 3 |- ran aleph (_ (om u. ran aleph)
11 fss 3626 . . 3 |- ((H:om-->ran aleph /\ ran aleph (_ (om u. ran aleph)) -> H:om-->(om u. ran aleph))
129, 10, 11mp2an 696 . 2 |- H:om-->(om u. ran aleph)
13 peano1 3144 . . 3 |- (/) e. om
143alephfplem1 4876 . . 3 |- (H` (/)) e. ran aleph
15 fveq2 3715 . . . . 5 |- (z = (/) -> (H` z) = (H` (/)))
1615eleq1d 1537 . . . 4 |- (z = (/) -> ((H` z) e. ran aleph <-> (H` (/)) e. ran aleph))
1716rcla4ev 1873 . . 3 |- (((/) e. om /\ (H` (/)) e. ran aleph) -> E.z e. om (H` z) e. ran aleph)
1813, 14, 17mp2an 696 . 2 |- E.z e. om (H` z) e. ran aleph
19 omex 4607 . . 3 |- om e. V
20 cardinfima 4871 . . 3 |- (om e. V -> ((H:om-->(om u. ran aleph) /\ E.z e. om (H` z) e. ran aleph) -> U.(H"om) e. ran aleph))
2119, 20ax-mp 7 . 2 |- ((H:om-->(om u. ran aleph) /\ E.z e. om (H` z) e. ran aleph) -> U.(H"om) e. ran aleph)
2212, 18, 21mp2an 696 1 |- U.(H"om) e. ran aleph
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  A.wral 1642  E.wrex 1643  Vcvv 1807   u. cun 2041   (_ wss 2043  (/)c0 2276  U.cuni 2498  {copab 2661  omcom 3126  ran crn 3166   |` cres 3167  "cima 3168   Fn wfn 3172  -->wf 3173  ` cfv 3177  reccrdg 3922  alephcale 4794
This theorem is referenced by:  alephfp 4880  alephfp2 4881
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605  ax-ac 4724
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-er 4251  df-en 4357  df-dom 4358  df-sdom 4359  df-card 4796  df-aleph 4797
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