Theorem wfrlem1 13957

Description: Lemma for well-founded recursion. The final item we are interested in is the union of acceptable functions B. This lemma just changes bound variables for later use.

Hypothesis

Ref	Expression
wfrlem1.1	|- 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))))}

Assertion

Ref	Expression
wfrlem1	|- B = {g | E.z(g Fn z /\ (z C_ A /\ A.w e. z Pred(R, A, w) C_ z) /\ A.w e. z (g` w) = (G` (g |` Pred(R, A, w))))}

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

Proof of Theorem wfrlem1

Step	Hyp	Ref	Expression
1		wfrlem1.1	. 2 |- 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))))}
2		fneq1 4503	. . . . . 6 |- (f = g -> (f Fn x <-> g Fn x))
3		fveq1 4680	. . . . . . . 8 |- (f = g -> (f` y) = (g` y))
4		reseq1 4218	. . . . . . . . 9 |- (f = g -> (f |` Pred(R, A, y)) = (g |` Pred(R, A, y)))
5	4	fveq2d 4685	. . . . . . . 8 |- (f = g -> (G` (f |` Pred(R, A, y))) = (G` (g |` Pred(R, A, y))))
6	3, 5	eqeq12d 1899	. . . . . . 7 |- (f = g -> ((f` y) = (G` (f |` Pred(R, A, y))) <-> (g` y) = (G` (g |` Pred(R, A, y)))))
7	6	ralbidv 2123	. . . . . 6 |- (f = g -> (A.y e. x (f` y) = (G` (f |` Pred(R, A, y))) <-> A.y e. x (g` y) = (G` (g |` Pred(R, A, y)))))
8	2, 7	3anbi13d 1170	. . . . 5 |- (f = g -> ((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)))) <-> (g Fn x /\ (x C_ A /\ A.y e. x Pred(R, A, y) C_ x) /\ A.y e. x (g` y) = (G` (g |` Pred(R, A, y))))))
9	8	exbidv 1657	. . . 4 |- (f = g -> (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)))) <-> E.x(g Fn x /\ (x C_ A /\ A.y e. x Pred(R, A, y) C_ x) /\ A.y e. x (g` y) = (G` (g |` Pred(R, A, y))))))
10		fneq2 4504	. . . . . 6 |- (x = z -> (g Fn x <-> g Fn z))
11		sseq1 2637	. . . . . . 7 |- (x = z -> (x C_ A <-> z C_ A))
12		sseq2 2639	. . . . . . . . 9 |- (x = z -> (Pred(R, A, y) C_ x <-> Pred(R, A, y) C_ z))
13	12	raleqbi1dv 2271	. . . . . . . 8 |- (x = z -> (A.y e. x Pred(R, A, y) C_ x <-> A.y e. z Pred(R, A, y) C_ z))
14		predeq3 13883	. . . . . . . . . 10 |- (y = w -> Pred(R, A, y) = Pred(R, A, w))
15	14	sseq1d 2644	. . . . . . . . 9 |- (y = w -> (Pred(R, A, y) C_ z <-> Pred(R, A, w) C_ z))
16	15	cbvralv 2280	. . . . . . . 8 |- (A.y e. z Pred(R, A, y) C_ z <-> A.w e. z Pred(R, A, w) C_ z)
17	13, 16	syl6bb 595	. . . . . . 7 |- (x = z -> (A.y e. x Pred(R, A, y) C_ x <-> A.w e. z Pred(R, A, w) C_ z))
18	11, 17	anbi12d 690	. . . . . 6 |- (x = z -> ((x C_ A /\ A.y e. x Pred(R, A, y) C_ x) <-> (z C_ A /\ A.w e. z Pred(R, A, w) C_ z)))
19		raleq 2266	. . . . . . 7 |- (x = z -> (A.y e. x (g` y) = (G` (g |` Pred(R, A, y))) <-> A.y e. z (g` y) = (G` (g |` Pred(R, A, y)))))
20		fveq2 4681	. . . . . . . . 9 |- (y = w -> (g` y) = (g` w))
21		reseq2 4219	. . . . . . . . . . 11 |- (Pred(R, A, y) = Pred(R, A, w) -> (g |` Pred(R, A, y)) = (g |` Pred(R, A, w)))
22	14, 21	syl 12	. . . . . . . . . 10 |- (y = w -> (g |` Pred(R, A, y)) = (g |` Pred(R, A, w)))
23	22	fveq2d 4685	. . . . . . . . 9 |- (y = w -> (G` (g |` Pred(R, A, y))) = (G` (g |` Pred(R, A, w))))
24	20, 23	eqeq12d 1899	. . . . . . . 8 |- (y = w -> ((g` y) = (G` (g |` Pred(R, A, y))) <-> (g` w) = (G` (g |` Pred(R, A, w)))))
25	24	cbvralv 2280	. . . . . . 7 |- (A.y e. z (g` y) = (G` (g |` Pred(R, A, y))) <-> A.w e. z (g` w) = (G` (g |` Pred(R, A, w))))
26	19, 25	syl6bb 595	. . . . . 6 |- (x = z -> (A.y e. x (g` y) = (G` (g |` Pred(R, A, y))) <-> A.w e. z (g` w) = (G` (g |` Pred(R, A, w)))))
27	10, 18, 26	3anbi123d 1168	. . . . 5 |- (x = z -> ((g Fn x /\ (x C_ A /\ A.y e. x Pred(R, A, y) C_ x) /\ A.y e. x (g` y) = (G` (g |` Pred(R, A, y)))) <-> (g Fn z /\ (z C_ A /\ A.w e. z Pred(R, A, w) C_ z) /\ A.w e. z (g` w) = (G` (g |` Pred(R, A, w))))))
28	27	cbvexv 1697	. . . 4 |- (E.x(g Fn x /\ (x C_ A /\ A.y e. x Pred(R, A, y) C_ x) /\ A.y e. x (g` y) = (G` (g |` Pred(R, A, y)))) <-> E.z(g Fn z /\ (z C_ A /\ A.w e. z Pred(R, A, w) C_ z) /\ A.w e. z (g` w) = (G` (g |` Pred(R, A, w)))))
29	9, 28	syl6bb 595	. . 3 |- (f = g -> (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)))) <-> E.z(g Fn z /\ (z C_ A /\ A.w e. z Pred(R, A, w) C_ z) /\ A.w e. z (g` w) = (G` (g |` Pred(R, A, w))))))
30	29	cbvabv 2420	. 2 |- {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))))} = {g | E.z(g Fn z /\ (z C_ A /\ A.w e. z Pred(R, A, w) C_ z) /\ A.w e. z (g` w) = (G` (g |` Pred(R, A, w))))}
31	1, 30	eqtri 1908	1 |- B = {g | E.z(g Fn z /\ (z C_ A /\ A.w e. z Pred(R, A, w) C_ z) /\ A.w e. z (g` w) = (G` (g |` Pred(R, A, w))))}

Syntax hints:   /\ wa 240   /\ w3a 858   = wceq 1298  E.wex 1326  {cab 1871  A.wral 2105   C_ wss 2593   |` cres 3988   Fn wfn 3993  ` cfv 3998  Predcpred 13879

This theorem is referenced by:  wfrlem2 13958  wfrlem3 13959  wfrlem4 13960  wfrlem9 13965

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-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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-v 2294  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-br 3339  df-opab 3396  df-id 3586  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  df-pred 13880

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