Theorem tfr1onlem3ag 6316

Description: Lemma for transfinite recursion. This lemma changes some bound variables in A (version of tfrlem3ag 6288 but for tfr1on 6329 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 5291	. . . 4 |- ( ( f = H /\ x = z ) -> ( f Fn x <-> H Fn z ) )
2		simpll 524	. . . . . . 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 5503	. . . . . 6 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( f ` y ) = ( H ` w ) )
5	2, 3	reseq12d 4892	. . . . . . 7 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( f |` y ) = ( H |` w ) )
6	5	fveq2d 5500	. . . . . 6 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( G ` ( f |` y ) ) = ( G ` ( H |` w ) ) )
7	4, 6	eqeq12d 2185	. . . . 5 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( ( f ` y ) = ( G ` ( f |` y ) ) <-> ( H ` w ) = ( G ` ( H |` w ) ) ) )
8		simplr 525	. . . . 5 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> x = z )
9	7, 8	cbvraldva2 2703	. . . 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 470	. . 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 2706	. 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 2877	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 1348   e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   |` cres 4613   Fn wfn 5193   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  tfr1onlem3  6317  tfr1onlemsucaccv  6320  tfr1onlembxssdm  6322  tfr1onlemres  6328