Theorem tfr1onlem3ag 6305

Description: Lemma for transfinite recursion. This lemma changes some bound variables in A (version of tfrlem3ag 6277 but for tfr1on 6318 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 5281	. . . 4 |- ( ( f = H /\ x = z ) -> ( f Fn x <-> H Fn z ) )
2		simpll 519	. . . . . . 7 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> f = H )
3		simpr 109	. . . . . . 7 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> y = w )
4	2, 3	fveq12d 5493	. . . . . 6 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( f ` y ) = ( H ` w ) )
5	2, 3	reseq12d 4885	. . . . . . 7 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( f |` y ) = ( H |` w ) )
6	5	fveq2d 5490	. . . . . 6 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( G ` ( f |` y ) ) = ( G ` ( H |` w ) ) )
7	4, 6	eqeq12d 2180	. . . . 5 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( ( f ` y ) = ( G ` ( f |` y ) ) <-> ( H ` w ) = ( G ` ( H |` w ) ) ) )
8		simplr 520	. . . . 5 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> x = z )
9	7, 8	cbvraldva2 2699	. . . 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 465	. . 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 2702	. 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 2873	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 103   <-> wb 104   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   |` cres 4606   Fn wfn 5183   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196

This theorem is referenced by:  tfr1onlem3  6306  tfr1onlemsucaccv  6309  tfr1onlembxssdm  6311  tfr1onlemres  6317