Theorem tfrlem3ag 6367

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 5351	. . . 4 |- ( ( f = G /\ x = z ) -> ( f Fn x <-> G Fn z ) )
2		simpll 527	. . . . . . 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 5565	. . . . . 6 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( f ` y ) = ( G ` w ) )
5	2, 3	reseq12d 4947	. . . . . . 7 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( f |` y ) = ( G |` w ) )
6	5	fveq2d 5562	. . . . . 6 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( F ` ( f |` y ) ) = ( F ` ( G |` w ) ) )
7	4, 6	eqeq12d 2211	. . . . 5 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( ( f ` y ) = ( F ` ( f |` y ) ) <-> ( G ` w ) = ( F ` ( G |` w ) ) ) )
8		simplr 528	. . . . 5 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> x = z )
9	7, 8	cbvraldva2 2736	. . . 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 473	. . 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 2739	. 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 2911	1 |- ( G e. _V -> ( G e. A <-> E. z e. On ( G Fn z /\ A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   {cab 2182   A.wral 2475   E.wrex 2476   _Vcvv 2763   Oncon0 4398   |` cres 4665   Fn wfn 5253   ` cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266

This theorem is referenced by:  tfrlemisucaccv  6383