Theorem imbrov2fvoveq 5943

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 5554	. . . 4 |- ( X = Y -> ( G ` X ) = ( G ` Y ) )
3	2	fvoveq1d 5940	. . 3 |- ( X = Y -> ( F ` ( ( G ` X ) .x. O ) ) = ( F ` ( ( G ` Y ) .x. O ) ) )
4	3	breq1d 4039	. 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 1364   class class class wbr 4029   ` cfv 5254  (class class class)co 5918

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-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by:  cncfco  14746  mulcncflem  14761  ivthinclemlopn  14790  ivthinclemuopn  14792  limcimolemlt  14818  eflt  14910