Theorem tfrlem3a 6207

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 5216	. . . 4 |- ( ( f = G /\ x = z ) -> ( f Fn x <-> G Fn z ) )
3		simpll 518	. . . . . . 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 5428	. . . . . 6 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( f ` y ) = ( G ` w ) )
6	3, 4	reseq12d 4820	. . . . . . 7 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( f |` y ) = ( G |` w ) )
7	6	fveq2d 5425	. . . . . 6 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> ( F ` ( f |` y ) ) = ( F ` ( G |` w ) ) )
8	5, 7	eqeq12d 2154	. . . . 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 274	. . . . 5 |- ( ( ( f = G /\ x = z ) /\ y = w ) -> x = z )
11	8, 10	cbvraldva2 2661	. . . 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 464	. . 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 2664	. 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 2832	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 1331   e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   _Vcvv 2686   Oncon0 4285   |` cres 4541   Fn wfn 5118   ` cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131

This theorem is referenced by:  tfrlem3  6208  tfrlem5  6211  tfrlemisucaccv  6222  tfrlemibxssdm  6224  tfrlemi14d  6230  tfrexlem  6231