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Mirrors > Home > NFE Home > Th. List > funss | GIF version |
Description: Subclass theorem for function predicate. (The proof was shortened by Mario Carneiro, 24-Jun-2014.) (Contributed by set.mm contributors, 16-Aug-1994.) (Revised by set.mm contributors, 24-Jun-2014.) |
Ref | Expression |
---|---|
funss | ⊢ (A ⊆ B → (Fun B → Fun A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coss1 4872 | . . . 4 ⊢ (A ⊆ B → (A ∘ ◡A) ⊆ (B ∘ ◡A)) | |
2 | cnvss 4885 | . . . . 5 ⊢ (A ⊆ B → ◡A ⊆ ◡B) | |
3 | coss2 4873 | . . . . 5 ⊢ (◡A ⊆ ◡B → (B ∘ ◡A) ⊆ (B ∘ ◡B)) | |
4 | 2, 3 | syl 15 | . . . 4 ⊢ (A ⊆ B → (B ∘ ◡A) ⊆ (B ∘ ◡B)) |
5 | 1, 4 | sstrd 3282 | . . 3 ⊢ (A ⊆ B → (A ∘ ◡A) ⊆ (B ∘ ◡B)) |
6 | sstr2 3279 | . . 3 ⊢ ((A ∘ ◡A) ⊆ (B ∘ ◡B) → ((B ∘ ◡B) ⊆ I → (A ∘ ◡A) ⊆ I )) | |
7 | 5, 6 | syl 15 | . 2 ⊢ (A ⊆ B → ((B ∘ ◡B) ⊆ I → (A ∘ ◡A) ⊆ I )) |
8 | df-fun 4789 | . 2 ⊢ (Fun B ↔ (B ∘ ◡B) ⊆ I ) | |
9 | df-fun 4789 | . 2 ⊢ (Fun A ↔ (A ∘ ◡A) ⊆ I ) | |
10 | 7, 8, 9 | 3imtr4g 261 | 1 ⊢ (A ⊆ B → (Fun B → Fun A)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3257 ∘ ccom 4721 I cid 4763 ◡ccnv 4771 Fun wfun 4775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-opab 4623 df-br 4640 df-co 4726 df-cnv 4785 df-fun 4789 |
This theorem is referenced by: funeq 5127 funopab4 5141 funres 5143 funin 5163 funres11 5164 foimacnv 5303 |
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