Theorem tfr1onlem3 6568

Description: Lemma for transfinite recursion. This lemma changes some bound variables in A (version of tfrlem3 6541 but for tfr1on 6580 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 2815	. . 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 6567	. . 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 2347	1 |- A = { g | E. z e. X ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) }

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   {cab 2218   A.wral 2520   E.wrex 2521   _Vcvv 2812   |` cres 4750   Fn wfn 5346   ` cfv 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359

This theorem is referenced by:  tfr1on  6580