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Mirrors > Home > MPE Home > Th. List > fcdmssb | Structured version Visualization version GIF version |
Description: A function is a function into a subset of its codomain if all of its values are elements of this subset. (Contributed by AV, 7-Feb-2021.) |
Ref | Expression |
---|---|
fcdmssb | ⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → (𝐹:𝐴⟶𝑊 ↔ 𝐹:𝐴⟶𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . 4 ⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) | |
2 | ffn 6710 | . . . 4 ⊢ (𝐹:𝐴⟶𝑊 → 𝐹 Fn 𝐴) | |
3 | 1, 2 | anim12ci 613 | . . 3 ⊢ (((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) ∧ 𝐹:𝐴⟶𝑊) → (𝐹 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉)) |
4 | ffnfv 7113 | . . 3 ⊢ (𝐹:𝐴⟶𝑉 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉)) | |
5 | 3, 4 | sylibr 233 | . 2 ⊢ (((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) ∧ 𝐹:𝐴⟶𝑊) → 𝐹:𝐴⟶𝑉) |
6 | simpl 482 | . . . 4 ⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → 𝑉 ⊆ 𝑊) | |
7 | 6 | anim1ci 615 | . . 3 ⊢ (((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) ∧ 𝐹:𝐴⟶𝑉) → (𝐹:𝐴⟶𝑉 ∧ 𝑉 ⊆ 𝑊)) |
8 | fss 6727 | . . 3 ⊢ ((𝐹:𝐴⟶𝑉 ∧ 𝑉 ⊆ 𝑊) → 𝐹:𝐴⟶𝑊) | |
9 | 7, 8 | syl 17 | . 2 ⊢ (((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) ∧ 𝐹:𝐴⟶𝑉) → 𝐹:𝐴⟶𝑊) |
10 | 5, 9 | impbida 798 | 1 ⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → (𝐹:𝐴⟶𝑊 ↔ 𝐹:𝐴⟶𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2098 ∀wral 3055 ⊆ wss 3943 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 |
This theorem is referenced by: wlkdlem1 29443 0prjspnrel 41929 |
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