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Theorem frnssb 6880
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
frnssb ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → (𝐹:𝐴𝑊𝐹:𝐴𝑉))
Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝑘,𝑉
Allowed substitution hint:   𝑊(𝑘)

Proof of Theorem frnssb
StepHypRef Expression
1 simpr 485 . . . 4 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉)
2 ffn 6510 . . . 4 (𝐹:𝐴𝑊𝐹 Fn 𝐴)
31, 2anim12ci 613 . . 3 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑊) → (𝐹 Fn 𝐴 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉))
4 ffnfv 6877 . . 3 (𝐹:𝐴𝑉 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉))
53, 4sylibr 235 . 2 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑊) → 𝐹:𝐴𝑉)
6 simpl 483 . . . 4 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → 𝑉𝑊)
76anim1ci 615 . . 3 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑉) → (𝐹:𝐴𝑉𝑉𝑊))
8 fss 6523 . . 3 ((𝐹:𝐴𝑉𝑉𝑊) → 𝐹:𝐴𝑊)
97, 8syl 17 . 2 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑉) → 𝐹:𝐴𝑊)
105, 9impbida 797 1 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → (𝐹:𝐴𝑊𝐹:𝐴𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2107  wral 3142  wss 3939   Fn wfn 6346  wf 6347  cfv 6351
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359
This theorem is referenced by:  wlkdlem1  27380  0prjspnrel  39136
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