Theorem tfrlem3ag 6580

Description: Lemma for transfinite recursion. This lemma just changes some bound variables in A for later use. (Contributed by Jim Kingdon, 5-Jul-2019.)

Hypothesis

Ref	Expression
tfrlem3.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }

Assertion

Ref	Expression
tfrlem3ag	|- ( G e. _V -> ( G e. A <-> E. z e. On ( G Fn z /\ A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) ) )

Distinct variable groups:   w, f, x, y, z, F   f, G, w, x, y, z

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

Proof of Theorem tfrlem3ag

Step	Hyp	Ref	Expression
1		fneq12 5474	. . . 4 |- ( ( f = G /\ x = z ) -> ( f Fn x <-> G Fn z ) )
2		simpll 531	. . . . . . 7 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> f = G )
3		simpr 110	. . . . . . 7 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> y = w )
4	2, 3	fveq12d 5702	. . . . . 6 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( f ` y ) = ( G ` w ) )
5	2, 3	reseq12d 5064	. . . . . . 7 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( f |` y ) = ( G |` w ) )
6	5	fveq2d 5699	. . . . . 6 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( F ` ( f |` y ) ) = ( F ` ( G |` w ) ) )
7	4, 6	eqeq12d 2253	. . . . 5 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( ( f ` y ) = ( F ` ( f |` y ) ) <-> ( G ` w ) = ( F ` ( G |` w ) ) ) )
8		simplr 533	. . . . 5 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> x = z )
9	7, 8	cbvraldva2 2793	. . . 4 |- ( ( f = G /\ x = z ) -> ( A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) <-> A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) )
10	1, 9	anbi12d 477	. . 3 |- ( ( f = G /\ x = z ) -> ( ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) <-> ( G Fn z /\ A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) ) )
11	10	cbvrexdva 2796	. 2 |- ( f = G -> ( E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) <-> E. z e. On ( G Fn z /\ A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) ) )
12		tfrlem3.1	. 2 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
13	11, 12	elab2g 2973	1 |- ( G e. _V -> ( G e. A <-> E. z e. On ( G Fn z /\ A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   _Vcvv 2821   Oncon0 4508   |` cres 4776   Fn wfn 5372   ` cfv 5377

This proof depends on 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 proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385

This theorem is used by:  tfrlemisucaccv  6596