Theorem frrlem1 31780

Description: Lemma for 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 Paul Chapman, 21-Apr-2012.)

Hypothesis

Ref	Expression
frrlem1.1	|- 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 ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) ) }

Assertion

Ref	Expression
frrlem1	|- 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 ) = ( 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

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

Proof of Theorem frrlem1

Step	Hyp	Ref	Expression
1		frrlem1.1	. 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 ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) ) }
2		fneq1 5979	. . . . . 6 |- ( f = g -> ( f Fn x <-> g Fn x ) )
3		fveq1 6190	. . . . . . . . 9 |- ( f = g -> ( f ` y ) = ( g ` y ) )
4		reseq1 5390	. . . . . . . . . 10 |- ( f = g -> ( f |` Pred ( R , A , y ) ) = ( g |` Pred ( R , A , y ) ) )
5	4	oveq2d 6666	. . . . . . . . 9 |- ( f = g -> ( y G ( f |` Pred ( R , A , y ) ) ) = ( y G ( g |` Pred ( R , A , y ) ) ) )
6	3, 5	eqeq12d 2637	. . . . . . . 8 |- ( f = g -> ( ( f ` y ) = ( y G ( f |` Pred ( R , A , y ) ) ) <-> ( g ` y ) = ( y G ( g |` Pred ( R , A , y ) ) ) ) )
7	6	ralbidv 2986	. . . . . . 7 |- ( f = g -> ( A. y e. x ( f ` y ) = ( y G ( f |` Pred ( R , A , y ) ) ) <-> A. y e. x ( g ` y ) = ( y G ( g |` Pred ( R , A , y ) ) ) ) )
8	7	3anbi3d 1405	. . . . . 6 |- ( f = g -> ( ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x /\ A. y e. x ( f ` y ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) <-> ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x /\ A. y e. x ( g ` y ) = ( y G ( g |` Pred ( R , A , y ) ) ) ) ) )
9	2, 8	anbi12d 747	. . . . 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 ) = ( 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 ) = ( y G ( g |` Pred ( R , A , y ) ) ) ) ) ) )
10	9	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 ) = ( 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 ) = ( y G ( g |` Pred ( R , A , y ) ) ) ) ) ) )
11		fneq2 5980	. . . . . 6 |- ( x = z -> ( g Fn x <-> g Fn z ) )
12		sseq1 3626	. . . . . . 7 |- ( x = z -> ( x C_ A <-> z C_ A ) )
13		sseq2 3627	. . . . . . . . 9 |- ( x = z -> ( Pred ( R , A , y ) C_ x <-> Pred ( R , A , y ) C_ z ) )
14	13	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 ) )
15		predeq3 5684	. . . . . . . . . 10 |- ( y = w -> Pred ( R , A , y ) = Pred ( R , A , w ) )
16	15	sseq1d 3632	. . . . . . . . 9 |- ( y = w -> ( Pred ( R , A , y ) C_ z <-> Pred ( R , A , w ) C_ z ) )
17	16	cbvralv 3171	. . . . . . . 8 |- ( A. y e. z Pred ( R , A , y ) C_ z <-> A. w e. z Pred ( R , A , w ) C_ z )
18	14, 17	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 ) )
19		raleq 3138	. . . . . . . 8 |- ( x = z -> ( A. y e. x ( g ` y ) = ( y G ( g |` Pred ( R , A , y ) ) ) <-> A. y e. z ( g ` y ) = ( y G ( g |` Pred ( R , A , y ) ) ) ) )
20		fveq2 6191	. . . . . . . . . 10 |- ( y = w -> ( g ` y ) = ( g ` w ) )
21		id 22	. . . . . . . . . . 11 |- ( y = w -> y = w )
22	15	reseq2d 5396	. . . . . . . . . . 11 |- ( y = w -> ( g |` Pred ( R , A , y ) ) = ( g |` Pred ( R , A , w ) ) )
23	21, 22	oveq12d 6668	. . . . . . . . . 10 |- ( y = w -> ( y G ( g |` Pred ( R , A , y ) ) ) = ( w G ( g |` Pred ( R , A , w ) ) ) )
24	20, 23	eqeq12d 2637	. . . . . . . . 9 |- ( y = w -> ( ( g ` y ) = ( y G ( g |` Pred ( R , A , y ) ) ) <-> ( g ` w ) = ( w G ( g |` Pred ( R , A , w ) ) ) ) )
25	24	cbvralv 3171	. . . . . . . 8 |- ( A. y e. z ( g ` y ) = ( y G ( g |` Pred ( R , A , y ) ) ) <-> A. w e. z ( g ` w ) = ( w G ( g |` Pred ( R , A , w ) ) ) )
26	19, 25	syl6bb 276	. . . . . . 7 |- ( x = z -> ( A. y e. x ( g ` y ) = ( y G ( g |` Pred ( R , A , y ) ) ) <-> A. w e. z ( g ` w ) = ( w G ( g |` Pred ( R , A , w ) ) ) ) )
27	12, 18, 26	3anbi123d 1399	. . . . . 6 |- ( x = z -> ( ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x /\ A. y e. x ( g ` y ) = ( y G ( g |` Pred ( R , A , y ) ) ) ) <-> ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z /\ A. w e. z ( g ` w ) = ( w G ( g |` Pred ( R , A , w ) ) ) ) ) )
28	11, 27	anbi12d 747	. . . . 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 ) = ( 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 ) = ( w G ( g |` Pred ( R , A , w ) ) ) ) ) ) )
29	28	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 ) = ( 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 ) = ( w G ( g |` Pred ( R , A , w ) ) ) ) ) )
30	10, 29	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 ) = ( 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 ) = ( w G ( g |` Pred ( R , A , w ) ) ) ) ) ) )
31	30	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 ) = ( 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 ) = ( w G ( g |` Pred ( R , A , w ) ) ) ) ) }
32	1, 31	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 ) = ( w G ( 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  (class class class)co 6650

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  df-ov 6653

This theorem is referenced by:  frrlem2  31781  frrlem3  31782  frrlem4  31783  frrlem5e  31788