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Mirrors > Home > MPE Home > Th. List > Mathboxes > funALTVss | Structured version Visualization version GIF version |
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 |
---|---|
funALTVss | ⊢ (𝐴 ⊆ 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cossss 35704 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵) | |
2 | sstr2 3967 | . . . 4 ⊢ ( ≀ 𝐴 ⊆ ≀ 𝐵 → ( ≀ 𝐵 ⊆ I → ≀ 𝐴 ⊆ I )) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ( ≀ 𝐵 ⊆ I → ≀ 𝐴 ⊆ I )) |
4 | relss 5649 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴)) | |
5 | 3, 4 | anim12d 610 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (( ≀ 𝐵 ⊆ I ∧ Rel 𝐵) → ( ≀ 𝐴 ⊆ I ∧ Rel 𝐴))) |
6 | dffunALTV2 35955 | . 2 ⊢ ( FunALTV 𝐵 ↔ ( ≀ 𝐵 ⊆ I ∧ Rel 𝐵)) | |
7 | dffunALTV2 35955 | . 2 ⊢ ( FunALTV 𝐴 ↔ ( ≀ 𝐴 ⊆ I ∧ Rel 𝐴)) | |
8 | 5, 6, 7 | 3imtr4g 298 | 1 ⊢ (𝐴 ⊆ 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ⊆ wss 3929 I cid 5452 Rel wrel 5553 ≀ ccoss 35487 FunALTV wfunALTV 35518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-coss 35693 df-cnvrefrel 35799 df-funALTV 35949 |
This theorem is referenced by: funALTVeq 35967 disjss 35998 |
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