Theorem tfr1onlem3ag 6598

Description: Lemma for transfinite recursion. This lemma changes some bound variables in A (version of tfrlem3ag 6570 but for tfr1on 6611 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 5469	. . . 4 |- ( ( f = H /\ x = z ) -> ( f Fn x <-> H Fn z ) )
2		simpll 531	. . . . . . 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 5697	. . . . . 6 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( f ` y ) = ( H ` w ) )
5	2, 3	reseq12d 5059	. . . . . . 7 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( f |` y ) = ( H |` w ) )
6	5	fveq2d 5694	. . . . . 6 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( G ` ( f |` y ) ) = ( G ` ( H |` w ) ) )
7	4, 6	eqeq12d 2253	. . . . 5 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> ( ( f ` y ) = ( G ` ( f |` y ) ) <-> ( H ` w ) = ( G ` ( H |` w ) ) ) )
8		simplr 533	. . . . 5 |- ( ( ( f = H /\ x = z ) /\ y = w ) -> x = z )
9	7, 8	cbvraldva2 2793	. . . 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 477	. . 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 2796	. 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 2973	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 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   |` cres 4771   Fn wfn 5367   ` cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380

This theorem is referenced by:  tfr1onlem3  6599  tfr1onlemsucaccv  6602  tfr1onlembxssdm  6604  tfr1onlemres  6610