Theorem mptfnd 42786

Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)

Hypotheses

Ref	Expression
mptfnd.1	⊢ Ⅎ𝑥𝐴
mptfnd.2	⊢ Ⅎ𝑥𝜑
mptfnd.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
mptfnd	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)

Proof of Theorem mptfnd

Step	Hyp	Ref	Expression
1		mptfnd.2	. . 3 ⊢ Ⅎ𝑥𝜑
2		mptfnd.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
3	2	ex 413	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉))
4		elex 3450	. . . 4 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
5	3, 4	syl6 35	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ V))
6	1, 5	ralrimi 3141	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
7		mptfnd.1	. . 3 ⊢ Ⅎ𝑥𝐴
8	7	mptfnf 6568	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
9	6, 8	sylib 217	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 396  Ⅎwnf 1786   ∈ wcel 2106  Ⅎwnfc 2887  ∀wral 3064  Vcvv 3432   ↦ cmpt 5157   Fn wfn 6428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-fun 6435  df-fn 6436

This theorem is referenced by:  smflimsuplem2  44354