Theorem wfrlem1 7414

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. (Contributed by Scott Fenton, 21-Apr-2011.)

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 ) = ( F ` ( 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 ) = ( F ` ( g |` Pred ( R , A , w ) ) ) ) }

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

Allowed substitution hints:   B( x, y, z, w, f, g)

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 ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) }
2		fneq1 5979	. . . . . 6 |- ( f = g -> ( f Fn x <-> g Fn x ) )
3		fveq1 6190	. . . . . . . 8 |- ( f = g -> ( f ` y ) = ( g ` y ) )
4		reseq1 5390	. . . . . . . . 9 |- ( f = g -> ( f |` Pred ( R , A , y ) ) = ( g |` Pred ( R , A , y ) ) )
5	4	fveq2d 6195	. . . . . . . 8 |- ( f = g -> ( F ` ( f |` Pred ( R , A , y ) ) ) = ( F ` ( g |` Pred ( R , A , y ) ) ) )
6	3, 5	eqeq12d 2637	. . . . . . 7 |- ( f = g -> ( ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) <-> ( g ` y ) = ( F ` ( g |` Pred ( R , A , y ) ) ) ) )
7	6	ralbidv 2986	. . . . . 6 |- ( f = g -> ( A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) <-> A. y e. x ( g ` y ) = ( F ` ( g |` Pred ( R , A , y ) ) ) ) )
8	2, 7	3anbi13d 1401	. . . . 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 ) = ( F ` ( 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 ) = ( F ` ( g |` Pred ( R , A , y ) ) ) ) ) )
9	8	exbidv 1850	. . . 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 ) = ( F ` ( 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 ) = ( F ` ( g |` Pred ( R , A , y ) ) ) ) ) )
10		fneq2 5980	. . . . . 6 |- ( x = z -> ( g Fn x <-> g Fn z ) )
11		sseq1 3626	. . . . . . 7 |- ( x = z -> ( x C_ A <-> z C_ A ) )
12		sseq2 3627	. . . . . . . . 9 |- ( x = z -> ( Pred ( R , A , y ) C_ x <-> Pred ( R , A , y ) C_ z ) )
13	12	raleqbi1dv 3146	. . . . . . . 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 5684	. . . . . . . . . 10 |- ( y = w -> Pred ( R , A , y ) = Pred ( R , A , w ) )
15	14	sseq1d 3632	. . . . . . . . 9 |- ( y = w -> ( Pred ( R , A , y ) C_ z <-> Pred ( R , A , w ) C_ z ) )
16	15	cbvralv 3171	. . . . . . . 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 276	. . . . . . 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 747	. . . . . 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 3138	. . . . . . 7 |- ( x = z -> ( A. y e. x ( g ` y ) = ( F ` ( g |` Pred ( R , A , y ) ) ) <-> A. y e. z ( g ` y ) = ( F ` ( g |` Pred ( R , A , y ) ) ) ) )
20		fveq2 6191	. . . . . . . . 9 |- ( y = w -> ( g ` y ) = ( g ` w ) )
21	14	reseq2d 5396	. . . . . . . . . 10 |- ( y = w -> ( g |` Pred ( R , A , y ) ) = ( g |` Pred ( R , A , w ) ) )
22	21	fveq2d 6195	. . . . . . . . 9 |- ( y = w -> ( F ` ( g |` Pred ( R , A , y ) ) ) = ( F ` ( g |` Pred ( R , A , w ) ) ) )
23	20, 22	eqeq12d 2637	. . . . . . . 8 |- ( y = w -> ( ( g ` y ) = ( F ` ( g |` Pred ( R , A , y ) ) ) <-> ( g ` w ) = ( F ` ( g |` Pred ( R , A , w ) ) ) ) )
24	23	cbvralv 3171	. . . . . . 7 |- ( A. y e. z ( g ` y ) = ( F ` ( g |` Pred ( R , A , y ) ) ) <-> A. w e. z ( g ` w ) = ( F ` ( g |` Pred ( R , A , w ) ) ) )
25	19, 24	syl6bb 276	. . . . . 6 |- ( x = z -> ( A. y e. x ( g ` y ) = ( F ` ( g |` Pred ( R , A , y ) ) ) <-> A. w e. z ( g ` w ) = ( F ` ( g |` Pred ( R , A , w ) ) ) ) )
26	10, 18, 25	3anbi123d 1399	. . . . 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 ) = ( F ` ( 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 ) = ( F ` ( g |` Pred ( R , A , w ) ) ) ) ) )
27	26	cbvexv 2275	. . . 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 ) = ( F ` ( 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 ) = ( F ` ( g |` Pred ( R , A , w ) ) ) ) )
28	9, 27	syl6bb 276	. . 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 ) = ( F ` ( 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 ) = ( F ` ( g |` Pred ( R , A , w ) ) ) ) ) )
29	28	cbvabv 2747	. 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 ) = ( F ` ( 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 ) = ( F ` ( g |` Pred ( R , A , w ) ) ) ) }
30	1, 29	eqtri 2644	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 ) = ( F ` ( g |` Pred ( R , A , w ) ) ) ) }

Syntax hints:   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   {cab 2608   A.wral 2912   C_ wss 3574   |` cres 5116   Predcpred 5679   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  wfrlem2  7415  wfrlem3  7416  wfrlem3a  7417  wfrlem4  7418  wfrdmcl  7423