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Mirrors > Home > ILE Home > Th. List > funss | 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 4370 | . . 3 ⊢ (A ⊆ B → (Rel B → Rel A)) | |
2 | coss1 4434 | . . . . 5 ⊢ (A ⊆ B → (A ∘ ◡A) ⊆ (B ∘ ◡A)) | |
3 | cnvss 4451 | . . . . . 6 ⊢ (A ⊆ B → ◡A ⊆ ◡B) | |
4 | coss2 4435 | . . . . . 6 ⊢ (◡A ⊆ ◡B → (B ∘ ◡A) ⊆ (B ∘ ◡B)) | |
5 | 3, 4 | syl 14 | . . . . 5 ⊢ (A ⊆ B → (B ∘ ◡A) ⊆ (B ∘ ◡B)) |
6 | 2, 5 | sstrd 2949 | . . . 4 ⊢ (A ⊆ B → (A ∘ ◡A) ⊆ (B ∘ ◡B)) |
7 | sstr2 2946 | . . . 4 ⊢ ((A ∘ ◡A) ⊆ (B ∘ ◡B) → ((B ∘ ◡B) ⊆ I → (A ∘ ◡A) ⊆ I )) | |
8 | 6, 7 | syl 14 | . . 3 ⊢ (A ⊆ B → ((B ∘ ◡B) ⊆ I → (A ∘ ◡A) ⊆ I )) |
9 | 1, 8 | anim12d 318 | . 2 ⊢ (A ⊆ B → ((Rel B ∧ (B ∘ ◡B) ⊆ I ) → (Rel A ∧ (A ∘ ◡A) ⊆ I ))) |
10 | df-fun 4847 | . 2 ⊢ (Fun B ↔ (Rel B ∧ (B ∘ ◡B) ⊆ I )) | |
11 | df-fun 4847 | . 2 ⊢ (Fun A ↔ (Rel A ∧ (A ∘ ◡A) ⊆ I )) | |
12 | 9, 10, 11 | 3imtr4g 194 | 1 ⊢ (A ⊆ B → (Fun B → Fun A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ⊆ wss 2911 I cid 4016 ◡ccnv 4287 ∘ ccom 4292 Rel wrel 4293 Fun wfun 4839 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-in 2918 df-ss 2925 df-br 3756 df-opab 3810 df-rel 4295 df-cnv 4296 df-co 4297 df-fun 4847 |
This theorem is referenced by: funeq 4864 funopab4 4880 funres 4884 fun0 4900 funcnvcnv 4901 funin 4913 funres11 4914 foimacnv 5087 tfrlemibfn 5883 |
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