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Theorem tfr1 3909
Description: Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47. We start with an arbitrary class G, normally a function, and define a class A of all "acceptable" functions. The final function we're interested in is the union F of them. F is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of F. In this first part we show that F is a function whose domain is all ordinal numbers.
Hypotheses
Ref Expression
tfr.1 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
tfr.2 |- F = U.A
Assertion
Ref Expression
tfr1 |- F Fn On
Distinct variable groups:   x,y,f,A   x,F,y,f   x,G,y,f
Proof of Theorem tfr1
StepHypRef Expression
1 df-fn 3183 . 2 |- (F Fn On <-> (Fun F /\ dom F = On))
2 tfr.1 . . 3 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
3 tfr.2 . . 3 |- F = U.A
42, 3tfrlem7 3902 . 2 |- Fun F
5 eqid 1468 . . 3 |- (F u. {<.dom F, (G` (F |` dom F))>.}) = (F u. {<.dom F, (G` (F |` dom F))>.})
62, 3, 5tfrlem13 3908 . 2 |- dom F = On
71, 4, 6mpbir2an 728 1 |- F Fn On
Colors of variables: wff set class
Syntax hints:   /\ wa 223   = wceq 953  {cab 1456  A.wral 1637  E.wrex 1638   u. cun 2035  {csn 2399  <.cop 2401  U.cuni 2493  Oncon0 2938  dom cdm 3160   |` cres 3162  Fun wfun 3166   Fn wfn 3167  ` cfv 3172
This theorem is referenced by:  tfr3 3911  rdgfnon 3924  numthlem 4755  zorn2lem2 4761  zorn2lem4 4763  zorn2lem6 4765
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-rep 2683  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-3or 774  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-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  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-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  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-fv 3188
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