Theorem tfrlem3-2d 6477

Description: Lemma for transfinite recursion which changes a bound variable (Contributed by Jim Kingdon, 2-Jul-2019.)

Hypothesis

Ref	Expression
tfrlem3-2d.1	|- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )

Assertion

Ref	Expression
tfrlem3-2d	|- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) )

Distinct variable group:   x, g, F

Allowed substitution hints:   ph( x, g)

Proof of Theorem tfrlem3-2d

Step	Hyp	Ref	Expression
1		tfrlem3-2d.1	. . 3 |- ( ph -> A. x ( Fun F /\ ( F ` x ) e. _V ) )
2		fveq2 5639	. . . . . 6 |- ( x = g -> ( F ` x ) = ( F ` g ) )
3	2	eleq1d 2300	. . . . 5 |- ( x = g -> ( ( F ` x ) e. _V <-> ( F ` g ) e. _V ) )
4	3	anbi2d 464	. . . 4 |- ( x = g -> ( ( Fun F /\ ( F ` x ) e. _V ) <-> ( Fun F /\ ( F ` g ) e. _V ) ) )
5	4	cbvalv 1966	. . 3 |- ( A. x ( Fun F /\ ( F ` x ) e. _V ) <-> A. g ( Fun F /\ ( F ` g ) e. _V ) )
6	1, 5	sylib 122	. 2 |- ( ph -> A. g ( Fun F /\ ( F ` g ) e. _V ) )
7	6	19.21bi 1606	1 |- ( ph -> ( Fun F /\ ( F ` g ) e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1395   e. wcel 2202   _Vcvv 2802   Fun wfun 5320   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334

This theorem is referenced by:  tfrlemisucfn  6489  tfrlemisucaccv  6490  tfrlemibxssdm  6492  tfrlemibfn  6493  tfrlemi14d  6498