Theorem tfr2 5133

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 4681	. . 3 |- (y = z -> (F` y) = (F` z))
2		reseq2 4219	. . . 4 |- (y = z -> (F |` y) = (F |` z))
3	2	fveq2d 4685	. . 3 |- (y = z -> (G` (F |` y)) = (G` (F |` z)))
4	1, 3	eqeq12d 1899	. 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 1884	. . . . 5 |- (F u. {<.dom F, (G` (F |` dom F))>.}) = (F u. {<.dom F, (G` (F |` dom F))>.})
8	5, 6, 7	tfrlem13 5131	. . . 4 |- dom F = On
9	8	eleq2i 1961	. . 3 |- (y e. dom F <-> y e. On)
10	5, 6	tfrlem9 5127	. . 3 |- (y e. dom F -> (F` y) = (G` (F |` y)))
11	9, 10	sylbir 218	. 2 |- (y e. On -> (F` y) = (G` (F |` y)))
12	4, 11	vtoclga 2352	1 |- (z e. On -> (F` z) = (G` (F |` z)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106   u. cun 2591  {csn 3044  <.cop 3046  U.cuni 3177  Oncon0 3657  dom cdm 3986   |` cres 3988   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  tfr3 5134  rdgval 5148  ordtypelem1 5684  ordtypelem6 5689  numthlem 5945  zorn2lem1 5950  ordtypelem1OLD 15375  ordtypelem6OLD 15380

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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