Theorem tfrlem3a 6137

Description: Lemma for transfinite recursion. Let A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in A for later use. (Contributed by NM, 9-Apr-1995.)

Hypotheses

Ref	Expression
tfrlem3.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
tfrlem3.2	|- G e. _V

Assertion

Ref	Expression
tfrlem3a	|- ( 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 tfrlem3a

Step	Hyp	Ref	Expression
1		tfrlem3.2	. 2 |- G e. _V
2		fneq12 5152	. . . 4 |- ( ( f = G /\ x = z ) -> ( f Fn x <-> G Fn z ) )
3		simpll 499	. . . . . . 7 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> f = G )
4		simpr 109	. . . . . . 7 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> y = w )
5	3, 4	fveq12d 5360	. . . . . 6 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( f ` y ) = ( G ` w ) )
6	3, 4	reseq12d 4756	. . . . . . 7 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( f |` y ) = ( G |` w ) )
7	6	fveq2d 5357	. . . . . 6 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( F ` ( f |` y ) ) = ( F ` ( G |` w ) ) )
8	5, 7	eqeq12d 2114	. . . . 5 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( ( f ` y ) = ( F ` ( f |` y ) ) <-> ( G ` w ) = ( F ` ( G |` w ) ) ) )
9		simpr 109	. . . . . 6 |- ( ( f = G /\ x = z ) -> x = z )
10	9	adantr 272	. . . . 5 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> x = z )
11	8, 10	cbvraldva2 2616	. . . 4 |- ( ( f = G /\ x = z ) -> ( A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) <-> A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) )
12	2, 11	anbi12d 460	. . 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 ) ) ) ) )
13	12	cbvrexdva 2619	. 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 ) ) ) ) )
14		tfrlem3.1	. 2 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
15	1, 13, 14	elab2 2785	1 |- ( G e. A <-> E. z e. On ( G Fn z /\ A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1299   e. wcel 1448   {cab 2086   A.wral 2375   E.wrex 2376   _Vcvv 2641   Oncon0 4223   |` cres 4479   Fn wfn 5054   ` cfv 5059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-res 4489  df-iota 5024  df-fun 5061  df-fn 5062  df-fv 5067

This theorem is referenced by:  tfrlem3  6138  tfrlem5  6141  tfrlemisucaccv  6152  tfrlemibxssdm  6154  tfrlemi14d  6160  tfrexlem  6161