Theorem tfr2 3925

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 3724	. . 3 |- (y = z -> (F` y) = (F` z))
2		reseq2 3369	. . . 4 |- (y = z -> (F |` y) = (F |` z))
3	2	fveq2d 3728	. . 3 |- (y = z -> (G` (F |` y)) = (G` (F |` z)))
4	1, 3	eqeq12d 1489	. 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 1475	. . . . 5 |- (F u. {<.dom F, (G` (F |` dom F))>.}) = (F u. {<.dom F, (G` (F |` dom F))>.})
8	5, 6, 7	tfrlem13 3923	. . . 4 |- dom F = On
9	8	eleq2i 1538	. . 3 |- (y e. dom F <-> y e. On)
10	5, 6	tfrlem9 3919	. . 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 1852	1 |- (z e. On -> (F` z) = (G` (F |` z)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  E.wrex 1646   u. cun 2045  {csn 2409  <.cop 2411  U.cuni 2503  Oncon0 2948  dom cdm 3170   |` cres 3172   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  tfr3 3926  rdgval 3940  numthlem 4783  zorn2lem1 4788

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

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