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Theorem frnssb 6346
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 477 . . . 4 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉)
2 ffn 6002 . . . 4 (𝐹:𝐴𝑊𝐹 Fn 𝐴)
31, 2anim12ci 590 . . 3 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑊) → (𝐹 Fn 𝐴 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉))
4 ffnfv 6343 . . 3 (𝐹:𝐴𝑉 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉))
53, 4sylibr 224 . 2 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑊) → 𝐹:𝐴𝑉)
6 simpl 473 . . . . 5 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → 𝑉𝑊)
76anim1i 591 . . . 4 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑉) → (𝑉𝑊𝐹:𝐴𝑉))
87ancomd 467 . . 3 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑉) → (𝐹:𝐴𝑉𝑉𝑊))
9 fss 6013 . . 3 ((𝐹:𝐴𝑉𝑉𝑊) → 𝐹:𝐴𝑊)
108, 9syl 17 . 2 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑉) → 𝐹:𝐴𝑊)
115, 10impbida 876 1 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → (𝐹:𝐴𝑊𝐹:𝐴𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wcel 1987  wral 2907  wss 3555   Fn wfn 5842  wf 5843  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855
This theorem is referenced by:  wlkdlem1  26448
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