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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 5129 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴)) | |
2 | coss1 5199 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐴)) | |
3 | cnvss 5216 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
4 | coss2 5200 | . . . . . 6 ⊢ (◡𝐴 ⊆ ◡𝐵 → (𝐵 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) |
6 | 2, 5 | sstrd 3578 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) |
7 | sstr2 3575 | . . . 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 584 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((Rel 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))) |
10 | df-fun 5806 | . 2 ⊢ (Fun 𝐵 ↔ (Rel 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I )) | |
11 | df-fun 5806 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) | |
12 | 9, 10, 11 | 3imtr4g 284 | 1 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ⊆ wss 3540 I cid 4948 ◡ccnv 5037 ∘ ccom 5042 Rel wrel 5043 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-in 3547 df-ss 3554 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-fun 5806 |
This theorem is referenced by: funeq 5823 funopab4 5839 funres 5843 fun0 5868 funcnvcnv 5870 funin 5879 funres11 5880 foimacnv 6067 funsssuppss 7208 strssd 15737 strle1 15800 xpsc0 16043 xpsc1 16044 pjpm 19871 frrlem5c 31030 subgrfun 40505 |
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