Theorem mptfnd 42675

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 412	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉))
4		elex 3440	. . . 4 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
5	3, 4	syl6 35	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ V))
6	1, 5	ralrimi 3139	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
7		mptfnd.1	. . 3 ⊢ Ⅎ𝑥𝐴
8	7	mptfnf 6552	. 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 395  Ⅎwnf 1787   ∈ wcel 2108  Ⅎwnfc 2886  ∀wral 3063  Vcvv 3422   ↦ cmpt 5153   Fn wfn 6413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-fun 6420  df-fn 6421

This theorem is referenced by:  smflimsuplem2  44241