Theorem tfr1onlem3 6609

Description: Lemma for transfinite recursion. This lemma changes some bound variables in A (version of tfrlem3 6582 but for tfr1on 6621 related lemmas). (Contributed by Jim Kingdon, 14-Mar-2022.)

Hypothesis

Ref	Expression
tfr1onlem3ag.1	|- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }

Assertion

Ref	Expression
tfr1onlem3	|- A = { g | E. z e. X ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) }

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

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

Proof of Theorem tfr1onlem3

Step	Hyp	Ref	Expression
1		vex 2824	. . 3 |- g e. _V
2		tfr1onlem3ag.1	. . . 4 |- A = { f | E. x e. X ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
3	2	tfr1onlem3ag 6608	. . 3 |- ( g e. _V -> ( g e. A <-> E. z e. X ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) ) )
4	1, 3	ax-mp 5	. 2 |- ( g e. A <-> E. z e. X ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) )
5	4	abbi2i 2353	1 |- A = { g | E. z e. X ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) }

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   _Vcvv 2821   |` cres 4776   Fn wfn 5372   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385

This theorem is used by:  tfr1on  6621