Theorem imbrov2fvoveq 5669

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 5299	. . . 4 |- ( X = Y -> ( G ` X ) = ( G ` Y ) )
3	2	fvoveq1d 5666	. . 3 |- ( X = Y -> ( F ` ( ( G ` X ) .x. O ) ) = ( F ` ( ( G ` Y ) .x. O ) ) )
4	3	breq1d 3853	. 2 |- ( X = Y -> ( ( F ` ( ( G ` X ) .x. O ) ) R A <-> ( F ` ( ( G ` Y ) .x. O ) ) R A ) )
5	1, 4	imbi12d 232	1 |- ( X = Y -> ( ( ph -> ( F ` ( ( G ` X ) .x. O ) ) R A ) <-> ( ps -> ( F ` ( ( G ` Y ) .x. O ) ) R A ) ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1289   class class class wbr 3843   ` cfv 5010  (class class class)co 5644

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-iota 4975  df-fv 5018  df-ov 5647

This theorem is referenced by:  cncfco  11530