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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 5620 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴)) | |
2 | coss1 5690 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐴)) | |
3 | cnvss 5707 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
4 | coss2 5691 | . . . . . 6 ⊢ (◡𝐴 ⊆ ◡𝐵 → (𝐵 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) |
6 | 2, 5 | sstrd 3925 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) |
7 | sstr2 3922 | . . . 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 611 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((Rel 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))) |
10 | df-fun 6326 | . 2 ⊢ (Fun 𝐵 ↔ (Rel 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I )) | |
11 | df-fun 6326 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) | |
12 | 9, 10, 11 | 3imtr4g 299 | 1 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ⊆ wss 3881 I cid 5424 ◡ccnv 5518 ∘ ccom 5523 Rel wrel 5524 Fun wfun 6318 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-br 5031 df-opab 5093 df-rel 5526 df-cnv 5527 df-co 5528 df-fun 6326 |
This theorem is referenced by: funeq 6344 funopab4 6361 funres 6366 fun0 6389 funcnvcnv 6391 funin 6400 funres11 6401 foimacnv 6607 funelss 7728 funsssuppss 7839 strssd 16525 strle1 16584 pjpm 20397 subgrfun 27071 setrecsss 45230 |
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