Theorem tfr2 3864

Description: Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47. Here we show that the function F has the property that for any function G whatsoever, the "next" value of F is G recursively applied to all "previous" values of F.

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
tfr2	|- (z e. On -> (F` z) = (G` (F |` z)))

Distinct variable groups:   x,y,f,A   x,F,y,f   x,G,y,f   y,z

Proof of Theorem tfr2

Step	Hyp	Ref	Expression
1		fveq2 3663	. . 3 |- (y = z -> (F` y) = (F` z))
2		reseq2 3320	. . . 4 |- (y = z -> (F |` y) = (F |` z))
3	2	fveq2d 3667	. . 3 |- (y = z -> (G` (F |` y)) = (G` (F |` z)))
4	1, 3	eqeq12d 1465	. 2 |- (y = z -> ((F` y) = (G` (F |` y)) <-> (F` z) = (G` (F |` z))))
5		tfr.1	. . . . 5 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
6		tfr.2	. . . . 5 |- F = U.A
7		eqid 1452	. . . . 5 |- (F u. {<.dom F, (G` (F |` dom F))>.}) = (F u. {<.dom F, (G` (F |` dom F))>.})
8	5, 6, 7	tfrlem13 3862	. . . 4 |- dom F = On
9	8	eleq2i 1514	. . 3 |- (y e. dom F <-> y e. On)
10	5, 6	tfrlem9 3858	. . 3 |- (y e. dom F -> (F` y) = (G` (F |` y)))
11	9, 10	sylbir 201	. 2 |- (y e. On -> (F` y) = (G` (F |` y)))
12	4, 11	vtoclga 1827	1 |- (z e. On -> (F` z) = (G` (F |` z)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 1099   e. wcel 1105  {cab 1440  A.wral 1621  E.wrex 1622   u. cun 2016  {csn 2380  <.cop 2382  U.cuni 2471  Oncon0 2911  dom cdm 3133   |` cres 3135   Fn wfn 3140  ` cfv 3145

This theorem is referenced by:  tfr3 3865  rdgval 3879  numthlem 4707  zorn2lem1 4712

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 773  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-rab 1628  df-v 1787  df-sbc 1913  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-tp 2386  df-op 2387  df-uni 2472  df-iun 2536  df-br 2588  df-opab 2635  df-tr 2649  df-eprel 2794  df-id 2797  df-po 2804  df-so 2814  df-fr 2880  df-we 2897  df-ord 2914  df-on 2915  df-suc 2917  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-fv 3161

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