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Theorem fcdmssb 7123
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 6717 . . . 4 (𝐹:𝐴𝑊𝐹 Fn 𝐴)
31, 2anim12ci 613 . . 3 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑊) → (𝐹 Fn 𝐴 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉))
4 ffnfv 7120 . . 3 (𝐹:𝐴𝑉 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉))
53, 4sylibr 233 . 2 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑊) → 𝐹:𝐴𝑉)
6 simpl 482 . . . 4 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → 𝑉𝑊)
76anim1ci 615 . . 3 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑉) → (𝐹:𝐴𝑉𝑉𝑊))
8 fss 6734 . . 3 ((𝐹:𝐴𝑉𝑉𝑊) → 𝐹:𝐴𝑊)
97, 8syl 17 . 2 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑉) → 𝐹:𝐴𝑊)
105, 9impbida 798 1 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → (𝐹:𝐴𝑊𝐹:𝐴𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2105  wral 3060  wss 3948   Fn wfn 6538  wf 6539  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551
This theorem is referenced by:  wlkdlem1  29372  0prjspnrel  41832
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