Theorem fcdmssb 7027

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 486	. . . 4 ⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉)
2		ffn 6630	. . . 4 ⊢ (𝐹:𝐴⟶𝑊 → 𝐹 Fn 𝐴)
3	1, 2	anim12ci 615	. . 3 ⊢ (((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) ∧ 𝐹:𝐴⟶𝑊) → (𝐹 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉))
4		ffnfv 7024	. . 3 ⊢ (𝐹:𝐴⟶𝑉 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉))
5	3, 4	sylibr 233	. 2 ⊢ (((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) ∧ 𝐹:𝐴⟶𝑊) → 𝐹:𝐴⟶𝑉)
6		simpl 484	. . . 4 ⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → 𝑉 ⊆ 𝑊)
7	6	anim1ci 617	. . 3 ⊢ (((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) ∧ 𝐹:𝐴⟶𝑉) → (𝐹:𝐴⟶𝑉 ∧ 𝑉 ⊆ 𝑊))
8		fss 6647	. . 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 799	1 ⊢ ((𝑉 ⊆ 𝑊 ∧ ∀𝑘 ∈ 𝐴 (𝐹‘𝑘) ∈ 𝑉) → (𝐹:𝐴⟶𝑊 ↔ 𝐹:𝐴⟶𝑉))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∈ wcel 2104  ∀wral 3062   ⊆ wss 3892   Fn wfn 6453  ⟶wf 6454  ‘cfv 6458

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fv 6466

This theorem is referenced by:  wlkdlem1  28095  0prjspnrel  40501