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Mirrors > Home > MPE Home > Th. List > funss | Structured version Visualization version GIF version |
Description: Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) |
Ref | Expression |
---|---|
funss | ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relss 5658 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴)) | |
2 | coss1 5728 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐴)) | |
3 | cnvss 5745 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
4 | coss2 5729 | . . . . . 6 ⊢ (◡𝐴 ⊆ ◡𝐵 → (𝐵 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) |
6 | 2, 5 | sstrd 3979 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) |
7 | sstr2 3976 | . . . 4 ⊢ ((𝐴 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵) → ((𝐵 ∘ ◡𝐵) ⊆ I → (𝐴 ∘ ◡𝐴) ⊆ I )) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝐵 ∘ ◡𝐵) ⊆ I → (𝐴 ∘ ◡𝐴) ⊆ I )) |
9 | 1, 8 | anim12d 610 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((Rel 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))) |
10 | df-fun 6359 | . 2 ⊢ (Fun 𝐵 ↔ (Rel 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I )) | |
11 | df-fun 6359 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) | |
12 | 9, 10, 11 | 3imtr4g 298 | 1 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ⊆ wss 3938 I cid 5461 ◡ccnv 5556 ∘ ccom 5561 Rel wrel 5562 Fun wfun 6351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-in 3945 df-ss 3954 df-br 5069 df-opab 5131 df-rel 5564 df-cnv 5565 df-co 5566 df-fun 6359 |
This theorem is referenced by: funeq 6377 funopab4 6394 funres 6399 fun0 6421 funcnvcnv 6423 funin 6432 funres11 6433 foimacnv 6634 funelss 7748 funsssuppss 7858 strssd 16535 strle1 16594 pjpm 20854 subgrfun 27065 setrecsss 44810 |
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