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Mirrors > Home > MPE Home > Th. List > frnssb | 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 |
---|---|
frnssb | ⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → (𝐹:𝐴⟶𝑊 ↔ 𝐹:𝐴⟶𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . . . 4 ⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) | |
2 | ffn 6508 | . . . 4 ⊢ (𝐹:𝐴⟶𝑊 → 𝐹 Fn 𝐴) | |
3 | 1, 2 | anim12ci 615 | . . 3 ⊢ (((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) ∧ 𝐹:𝐴⟶𝑊) → (𝐹 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉)) |
4 | ffnfv 6876 | . . 3 ⊢ (𝐹:𝐴⟶𝑉 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉)) | |
5 | 3, 4 | sylibr 236 | . 2 ⊢ (((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) ∧ 𝐹:𝐴⟶𝑊) → 𝐹:𝐴⟶𝑉) |
6 | simpl 485 | . . . 4 ⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → 𝑉 ⊆ 𝑊) | |
7 | 6 | anim1ci 617 | . . 3 ⊢ (((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) ∧ 𝐹:𝐴⟶𝑉) → (𝐹:𝐴⟶𝑉 ∧ 𝑉 ⊆ 𝑊)) |
8 | fss 6521 | . . 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 799 | 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 208 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 |
This theorem is referenced by: wlkdlem1 27458 0prjspnrel 39262 |
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