Description: Subclass theorem for function. (Contributed by NM, 16-Aug-1994) (Proof shortened by Mario Carneiro, 24-Jun-2014) (Revised by Peter Mazsa, 22-Sep-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | funALTVss | |- ( A C_ B -> ( FunALTV B -> FunALTV A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cossss | |- ( A C_ B -> ,~ A C_ ,~ B ) |
|
| 2 | sstr2 | |- ( ,~ A C_ ,~ B -> ( ,~ B C_ _I -> ,~ A C_ _I ) ) |
|
| 3 | 1 2 | syl | |- ( A C_ B -> ( ,~ B C_ _I -> ,~ A C_ _I ) ) |
| 4 | relss | |- ( A C_ B -> ( Rel B -> Rel A ) ) |
|
| 5 | 3 4 | anim12d | |- ( A C_ B -> ( ( ,~ B C_ _I /\ Rel B ) -> ( ,~ A C_ _I /\ Rel A ) ) ) |
| 6 | dffunALTV2 | |- ( FunALTV B <-> ( ,~ B C_ _I /\ Rel B ) ) |
|
| 7 | dffunALTV2 | |- ( FunALTV A <-> ( ,~ A C_ _I /\ Rel A ) ) |
|
| 8 | 5 6 7 | 3imtr4g | |- ( A C_ B -> ( FunALTV B -> FunALTV A ) ) |