Theorem tfr1onlem3 6235

Description: Lemma for transfinite recursion. This lemma changes some bound variables in A (version of tfrlem3 6208 but for tfr1on 6247 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 2689	. . 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 6234	. . 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 2254	1 |- A = { g | E. z e. X ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) ) }

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   _Vcvv 2686   |` 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:  tfr1on  6247