Theorem tfrlem3 6315

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.)

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
tfrlem3	|- A = { g | E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) }

Distinct variable groups:   A, g   f, g, w, x, y, z, F

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

Proof of Theorem tfrlem3

Step	Hyp	Ref	Expression
1		tfrlem3.1	. . 3 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
2		vex 2742	. . 3 |- g e. _V
3	1, 2	tfrlem3a 6314	. 2 |- ( g e. A <-> E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )
4	3	abbi2i 2292	1 |- A = { g | E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) }

Syntax hints:   /\ wa 104   = wceq 1353   {cab 2163   A.wral 2455   E.wrex 2456   Oncon0 4365   |` cres 4630   Fn wfn 5213   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by:  tfrlem4  6317  tfrlem8  6322  tfrlemi1  6336  tfrexlem  6338  tfri1d  6339  tfrex  6372