Theorem mptfnd 46073

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 418	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉))
4		elex 3474	. . . 4 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
5	3, 4	syl6 36	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ V))
6	1, 5	ralrimi 3262	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
7		mptfnd.1	. . 3 ⊢ Ⅎ𝑥𝐴
8	7	mptfnf 6671	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
9	6, 8	sylib 221	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  Ⅎwnf 1816   ∈ wcel 2145  Ⅎwnfc 2909  ∀wral 3078  Vcvv 3453   ↦ cmpt 5190   Fn wfn 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-fun 6539  df-fn 6540

This theorem is used by:  smflimsuplem2  47651