Theorem imbrov2fvoveq 5969

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 5576	. . . 4 |- ( X = Y -> ( G ` X ) = ( G ` Y ) )
3	2	fvoveq1d 5966	. . 3 |- ( X = Y -> ( F ` ( ( G ` X ) .x. O ) ) = ( F ` ( ( G ` Y ) .x. O ) ) )
4	3	breq1d 4054	. 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 1373   class class class wbr 4044   ` cfv 5271  (class class class)co 5944

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  cncfco  15063  mulcncflem  15079  ivthinclemlopn  15108  ivthinclemuopn  15110  limcimolemlt  15136  eflt  15247