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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 4691 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴)) | |
2 | coss1 4759 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐴)) | |
3 | cnvss 4777 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | |
4 | coss2 4760 | . . . . . 6 ⊢ (◡𝐴 ⊆ ◡𝐵 → (𝐵 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) | |
5 | 3, 4 | syl 14 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) |
6 | 2, 5 | sstrd 3152 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ ◡𝐴) ⊆ (𝐵 ∘ ◡𝐵)) |
7 | sstr2 3149 | . . . 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 333 | . 2 ⊢ (𝐴 ⊆ 𝐵 → ((Rel 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I ) → (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I ))) |
10 | df-fun 5190 | . 2 ⊢ (Fun 𝐵 ↔ (Rel 𝐵 ∧ (𝐵 ∘ ◡𝐵) ⊆ I )) | |
11 | df-fun 5190 | . 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴 ∘ ◡𝐴) ⊆ I )) | |
12 | 9, 10, 11 | 3imtr4g 204 | 1 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ⊆ wss 3116 I cid 4266 ◡ccnv 4603 ∘ ccom 4608 Rel wrel 4609 Fun wfun 5182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147 |
This theorem depends on definitions: df-bi 116 df-nf 1449 df-sb 1751 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-in 3122 df-ss 3129 df-br 3983 df-opab 4044 df-rel 4611 df-cnv 4612 df-co 4613 df-fun 5190 |
This theorem is referenced by: funeq 5208 funopab4 5225 funres 5229 fun0 5246 funcnvcnv 5247 funin 5259 funres11 5260 foimacnv 5450 tfrlemibfn 6296 tfr1onlembfn 6312 tfrcllembfn 6325 strslssd 12440 strle1g 12485 |
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