Theorem tfr1onlem3 6391

Description: Lemma for transfinite recursion. This lemma changes some bound variables in A (version of tfrlem3 6364 but for tfr1on 6403 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 2763	. . 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 6390	. . 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 2308	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 1364   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   _Vcvv 2760   |` cres 4661   Fn wfn 5249   ` cfv 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by:  tfr1on  6403