Theorem wfr2 13974

Description: The Principle of Well-Founded Recursion, part 2 of 3. Next, we show that the value of F at any z e. A is G recursively applied to all "previous" values of F.

Hypotheses

Ref	Expression
wfr2.1	|- R We A
wfr2.2	|- A.x e. A Pred(R, A, x) e. _V
wfr2.3	|- B = {f | E.x(f Fn x /\ (x C_ A /\ A.y e. x Pred(R, A, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred(R, A, y))))}
wfr2.4	|- F = U.B

Assertion

Ref	Expression
wfr2	|- (z e. A -> (F` z) = (G` (F |` Pred(R, A, z))))

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

Proof of Theorem wfr2

Step	Hyp	Ref	Expression
1		fveq2 4681	. . 3 |- (y = z -> (F` y) = (F` z))
2		predeq3 13883	. . . . 5 |- (y = z -> Pred(R, A, y) = Pred(R, A, z))
3		reseq2 4219	. . . . 5 |- (Pred(R, A, y) = Pred(R, A, z) -> (F |` Pred(R, A, y)) = (F |` Pred(R, A, z)))
4	2, 3	syl 12	. . . 4 |- (y = z -> (F |` Pred(R, A, y)) = (F |` Pred(R, A, z)))
5	4	fveq2d 4685	. . 3 |- (y = z -> (G` (F |` Pred(R, A, y))) = (G` (F |` Pred(R, A, z))))
6	1, 5	eqeq12d 1899	. 2 |- (y = z -> ((F` y) = (G` (F |` Pred(R, A, y))) <-> (F` z) = (G` (F |` Pred(R, A, z)))))
7		wfr2.1	. . . . 5 |- R We A
8		wfr2.2	. . . . 5 |- A.x e. A Pred(R, A, x) e. _V
9		wfr2.3	. . . . 5 |- B = {f | E.x(f Fn x /\ (x C_ A /\ A.y e. x Pred(R, A, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred(R, A, y))))}
10		wfr2.4	. . . . 5 |- F = U.B
11		eqid 1884	. . . . 5 |- (F u. {<.w, (G` (F |` Pred(R, A, w)))>.}) = (F u. {<.w, (G` (F |` Pred(R, A, w)))>.})
12	7, 8, 9, 10, 11	wfrlem16 13972	. . . 4 |- dom F = A
13	12	eleq2i 1961	. . 3 |- (y e. dom F <-> y e. A)
14	7, 8, 9, 10	wfrlem12 13968	. . 3 |- (y e. dom F -> (F` y) = (G` (F |` Pred(R, A, y))))
15	13, 14	sylbir 218	. 2 |- (y e. A -> (F` y) = (G` (F |` Pred(R, A, y))))
16	6, 15	vtoclga 2352	1 |- (z e. A -> (F` z) = (G` (F |` Pred(R, A, z))))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  _Vcvv 2292   u. cun 2591   C_ wss 2593  {csn 3044  <.cop 3046  U.cuni 3177   We wwe 3624  dom cdm 3986   |` cres 3988   Fn wfn 3993  ` cfv 3998  Predcpred 13879

This theorem is referenced by:  wfr3 13975  tfr2ALT 13978

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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-pred 13880

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