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| 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 5791 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴)) | |
| 2 | coss1 5866 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐴)) | |
| 3 | cnvss 5883 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
| 4 | coss2 5867 | . . . . . 6 ⊢ (◡𝐴 ⊆ ◡𝐵 → (𝐵 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) | |
| 5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) | 
| 6 | 2, 5 | sstrd 3994 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) | 
| 7 | sstr2 3990 | . . . 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 609 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((Rel 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))) | 
| 10 | df-fun 6563 | . 2 ⊢ (Fun 𝐵 ↔ (Rel 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I )) | |
| 11 | df-fun 6563 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) | |
| 12 | 9, 10, 11 | 3imtr4g 296 | 1 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ⊆ wss 3951 I cid 5577 ◡ccnv 5684 ∘ ccom 5689 Rel wrel 5690 Fun wfun 6555 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ss 3968 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-fun 6563 | 
| This theorem is referenced by: funeq 6586 funopab4 6603 funres 6608 fun0 6631 funcnvcnv 6633 funin 6642 funres11 6643 foimacnv 6865 funelss 8072 funsssuppss 8215 fsuppss 9423 strle1 17195 strssd 17242 pjpm 21728 subgrfun 29298 setrecsss 49220 | 
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