Theorem imbrov2fvoveq 5890

Description: Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.)

Hypothesis

Ref	Expression
imbrov2fvoveq.1	|- ( X = Y -> ( ph <-> ps ) )

Assertion

Ref	Expression
imbrov2fvoveq	|- ( X = Y -> ( ( ph -> ( F ` ( ( G ` X ) .x. O ) ) R A ) <-> ( ps -> ( F ` ( ( G ` Y ) .x. O ) ) R A ) ) )

Proof of Theorem imbrov2fvoveq

Step	Hyp	Ref	Expression
1		imbrov2fvoveq.1	. 2 |- ( X = Y -> ( ph <-> ps ) )
2		fveq2 5507	. . . 4 |- ( X = Y -> ( G ` X ) = ( G ` Y ) )
3	2	fvoveq1d 5887	. . 3 |- ( X = Y -> ( F ` ( ( G ` X ) .x. O ) ) = ( F ` ( ( G ` Y ) .x. O ) ) )
4	3	breq1d 4008	. 2 |- ( X = Y -> ( ( F ` ( ( G ` X ) .x. O ) ) R A <-> ( F ` ( ( G ` Y ) .x. O ) ) R A ) )
5	1, 4	imbi12d 234	1 |- ( X = Y -> ( ( ph -> ( F ` ( ( G ` X ) .x. O ) ) R A ) <-> ( ps -> ( F ` ( ( G ` Y ) .x. O ) ) R A ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   class class class wbr 3998   ` cfv 5208  (class class class)co 5865

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-iota 5170  df-fv 5216  df-ov 5868

This theorem is referenced by:  cncfco  13647  mulcncflem  13659  ivthinclemlopn  13683  ivthinclemuopn  13685  limcimolemlt  13702  eflt  13765