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Theorem map0e 4326
Description: Set exponentiation with an empty exponent (ordinal number 0) is ordinal number 1. Exercise 4.42(a) of [Mendelson] p. 255.
Hypothesis
Ref Expression
map0e.1 |- A e. V
Assertion
Ref Expression
map0e |- (A ^m (/)) = 1o
Proof of Theorem map0e
StepHypRef Expression
1 fn0 3591 . . . . . 6 |- (f Fn (/) <-> f = (/))
21anbi1i 480 . . . . 5 |- ((f Fn (/) /\ ran f (_ A) <-> (f = (/) /\ ran f (_ A))
3 df-f 3184 . . . . 5 |- (f:(/)-->A <-> (f Fn (/) /\ ran f (_ A))
4 0ss 2291 . . . . . . 7 |- (/) (_ A
5 rneq 3328 . . . . . . . . 9 |- (f = (/) -> ran f = ran (/))
6 rn0 3341 . . . . . . . . 9 |- ran (/) = (/)
75, 6syl6eq 1515 . . . . . . . 8 |- (f = (/) -> ran f = (/))
87sseq1d 2078 . . . . . . 7 |- (f = (/) -> (ran f (_ A <-> (/) (_ A))
94, 8mpbiri 194 . . . . . 6 |- (f = (/) -> ran f (_ A)
109pm4.71i 635 . . . . 5 |- (f = (/) <-> (f = (/) /\ ran f (_ A))
112, 3, 103bitr4 183 . . . 4 |- (f:(/)-->A <-> f = (/))
1211abbii 1567 . . 3 |- {f | f:(/)-->A} = {f | f = (/)}
13 map0e.1 . . . 4 |- A e. V
14 0ex 2701 . . . 4 |- (/) e. V
1513, 14mapval 4316 . . 3 |- (A ^m (/)) = {f | f:(/)-->A}
16 df-sn 2402 . . 3 |- {(/)} = {f | f = (/)}
1712, 15, 163eqtr4 1497 . 2 |- (A ^m (/)) = {(/)}
18 df1o2 4124 . 2 |- 1o = {(/)}
1917, 18eqtr4 1490 1 |- (A ^m (/)) = 1o
Colors of variables: wff set class
Syntax hints:   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456  Vcvv 1802   (_ wss 2037  (/)c0 2270  {csn 2399  ran crn 3161   Fn wfn 3167  -->wf 3168  (class class class)co 3948  1oc1o 4112   ^m cm 4306
This theorem is referenced by:  map0 4328  infmap2 7523
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-oprab 3951  df-1o 4117  df-map 4308
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