Theorem tfrlem3a 6475

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 5423	. . . 4 |- ( ( f = G /\ x = z ) -> ( f Fn x <-> G Fn z ) )
3		simpll 527	. . . . . . 7 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> f = G )
4		simpr 110	. . . . . . 7 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> y = w )
5	3, 4	fveq12d 5646	. . . . . 6 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( f ` y ) = ( G ` w ) )
6	3, 4	reseq12d 5014	. . . . . . 7 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( f |` y ) = ( G |` w ) )
7	6	fveq2d 5643	. . . . . 6 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( F ` ( f |` y ) ) = ( F ` ( G |` w ) ) )
8	5, 7	eqeq12d 2246	. . . . 5 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( ( f ` y ) = ( F ` ( f |` y ) ) <-> ( G ` w ) = ( F ` ( G |` w ) ) ) )
9		simpr 110	. . . . . 6 |- ( ( f = G /\ x = z ) -> x = z )
10	9	adantr 276	. . . . 5 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> x = z )
11	8, 10	cbvraldva2 2774	. . . 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 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 ) ) ) ) )
13	12	cbvrexdva 2777	. 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 2954	1 |- ( G e. A <-> E. z e. On ( G Fn z /\ A. w e. z ( G ` w ) = ( F ` ( G |` w ) ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   {cab 2217   A.wral 2510   E.wrex 2511   _Vcvv 2802   Oncon0 4460   |` cres 4727   Fn wfn 5321   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by:  tfrlem3  6476  tfrlem5  6479  tfrlemisucaccv  6490  tfrlemibxssdm  6492  tfrlemi14d  6498  tfrexlem  6499