Theorem tfr1onlem3ag 6483

Description: Lemma for transfinite recursion. This lemma changes some bound variables in A (version of tfrlem3ag 6455 but for tfr1on 6496 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 5414	. . . 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 5634	. . . . . 6 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( f ` y ) = ( H ` w ) )
5	2, 3	reseq12d 5006	. . . . . . 7 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( f |` y ) = ( H |` w ) )
6	5	fveq2d 5631	. . . . . 6 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( G ` ( f |` y ) ) = ( G ` ( H |` w ) ) )
7	4, 6	eqeq12d 2244	. . . . 5 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( ( f ` y ) = ( G ` ( f |` y ) ) <-> ( H ` w ) = ( G ` ( H |` w ) ) ) )
8		simplr 528	. . . . 5 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> x = z )
9	7, 8	cbvraldva2 2772	. . . 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 2775	. 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 2950	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 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   |` cres 4721   Fn wfn 5313   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by:  tfr1onlem3  6484  tfr1onlemsucaccv  6487  tfr1onlembxssdm  6489  tfr1onlemres  6495