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Theorem fcdmssb 7142
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.)
Assertion
Ref Expression
fcdmssb ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → (𝐹:𝐴𝑊𝐹:𝐴𝑉))
Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝑘,𝑉
Allowed substitution hint:   𝑊(𝑘)

Proof of Theorem fcdmssb
StepHypRef Expression
1 simpr 484 . . . 4 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉)
2 ffn 6736 . . . 4 (𝐹:𝐴𝑊𝐹 Fn 𝐴)
31, 2anim12ci 614 . . 3 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑊) → (𝐹 Fn 𝐴 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉))
4 ffnfv 7139 . . 3 (𝐹:𝐴𝑉 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉))
53, 4sylibr 234 . 2 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑊) → 𝐹:𝐴𝑉)
6 simpl 482 . . . 4 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → 𝑉𝑊)
76anim1ci 616 . . 3 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑉) → (𝐹:𝐴𝑉𝑉𝑊))
8 fss 6752 . . 3 ((𝐹:𝐴𝑉𝑉𝑊) → 𝐹:𝐴𝑊)
97, 8syl 17 . 2 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑉) → 𝐹:𝐴𝑊)
105, 9impbida 801 1 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → (𝐹:𝐴𝑊𝐹:𝐴𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  wral 3061  wss 3951   Fn wfn 6556  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  wlkdlem1  29700  0prjspnrel  42637
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