Theorem fcdmssb 7067

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	⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → (𝐹:𝐴⟶𝑊 ↔ 𝐹:𝐴⟶𝑉))

Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝑘,𝑉

Allowed substitution hint:   𝑊(𝑘)

Proof of Theorem fcdmssb

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉)
2		ffn 6662	. . . 4 ⊢ (𝐹:𝐴⟶𝑊 → 𝐹 Fn 𝐴)
3	1, 2	anim12ci 614	. . 3 ⊢ (((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) ∧ 𝐹:𝐴⟶𝑊) → (𝐹 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉))
4		ffnfv 7064	. . 3 ⊢ (𝐹:𝐴⟶𝑉 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉))
5	3, 4	sylibr 234	. 2 ⊢ (((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) ∧ 𝐹:𝐴⟶𝑊) → 𝐹:𝐴⟶𝑉)
6		simpl 482	. . . 4 ⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → 𝑉 ⊆ 𝑊)
7	6	anim1ci 616	. . 3 ⊢ (((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) ∧ 𝐹:𝐴⟶𝑉) → (𝐹:𝐴⟶𝑉 ∧ 𝑉 ⊆ 𝑊))
8		fss 6678	. . 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 800	1 ⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → (𝐹:𝐴⟶𝑊 ↔ 𝐹:𝐴⟶𝑉))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113  ∀wral 3051   ⊆ wss 3901   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  wlkdlem1  29754  0prjspnrel  42870