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