Theorem tfrlem3 7474

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 3203	. . 3 |- g e. _V
3	1, 2	tfrlem3a 7473	. 2 |- ( g e. A <-> E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) )
4	3	abbi2i 2738	1 |- A = { g | E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( F ` ( g |` w ) ) ) }

Syntax hints:   /\ wa 384   = wceq 1483   {cab 2608   A.wral 2912   E.wrex 2913   |` cres 5116   Oncon0 5723   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  tfrlem4  7475  tfrlem8  7480  rdglem1  7511