Theorem tfr1onlem3ag 6502

Description: Lemma for transfinite recursion. This lemma changes some bound variables in A (version of tfrlem3ag 6474 but for tfr1on 6515 related lemmas). (Contributed by Jim Kingdon, 13-Mar-2022.)

Hypothesis

Ref	Expression
tfr1onlem3ag.1	|- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }

Assertion

Ref	Expression
tfr1onlem3ag	|- ( H e. V -> ( H e. A <-> E. z e. X ( H Fn z /\ A. w e. z ( H ` w ) = ( G ` ( H |` w ) ) ) ) )

Distinct variable groups:   f, G, w, x, y, z   f, H, w, x, y, z   f, X, x, z

Allowed substitution hints:   A( x, y, z, w, f)   V( x, y, z, w, f)   X( y, w)

Proof of Theorem tfr1onlem3ag

Step	Hyp	Ref	Expression
1		fneq12 5423	. . . 4 |- ( ( f = H /\ x = z ) -> ( f Fn x <-> H Fn z ) )
2		simpll 527	. . . . . . 7 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> f = H )
3		simpr 110	. . . . . . 7 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> y = w )
4	2, 3	fveq12d 5646	. . . . . 6 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( f ` y ) = ( H ` w ) )
5	2, 3	reseq12d 5014	. . . . . . 7 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( f |` y ) = ( H |` w ) )
6	5	fveq2d 5643	. . . . . 6 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( G ` ( f |` y ) ) = ( G ` ( H |` w ) ) )
7	4, 6	eqeq12d 2246	. . . . 5 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( ( f ` y ) = ( G ` ( f |` y ) ) <-> ( H ` w ) = ( G ` ( H |` w ) ) ) )
8		simplr 529	. . . . 5 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> x = z )
9	7, 8	cbvraldva2 2774	. . . 4 |- ( ( f = H /\ x = z ) -> ( A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) <-> A. w e. z ( H ` w ) = ( G ` ( H |` w ) ) ) )
10	1, 9	anbi12d 473	. . 3 |- ( ( f = H /\ x = z ) -> ( ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) <-> ( H Fn z /\ A. w e. z ( H ` w ) = ( G ` ( H |` w ) ) ) ) )
11	10	cbvrexdva 2777	. 2 |- ( f = H -> ( E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) <-> E. z e. X ( H Fn z /\ A. w e. z ( H ` w ) = ( G ` ( H |` w ) ) ) ) )
12		tfr1onlem3ag.1	. 2 |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
13	11, 12	elab2g 2953	1 |- ( H e. V -> ( H e. A <-> E. z e. X ( H Fn z /\ A. w e. z ( H ` w ) = ( G ` ( H |` w ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   {cab 2217   A.wral 2510   E.wrex 2511   |` cres 4727   Fn wfn 5321   ` cfv 5326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by:  tfr1onlem3  6503  tfr1onlemsucaccv  6506  tfr1onlembxssdm  6508  tfr1onlemres  6514