Theorem mptfnd 45808

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 416	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉))
4		elex 3476	. . . 4 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
5	3, 4	syl6 35	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ V))
6	1, 5	ralrimi 3261	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
7		mptfnd.1	. . 3 ⊢ Ⅎ𝑥𝐴
8	7	mptfnf 6656	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
9	6, 8	sylib 220	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 399  Ⅎwnf 1804   ∈ wcel 2143  Ⅎwnfc 2910  ∀wral 3077  Vcvv 3455   ↦ cmpt 5182   Fn wfn 6516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-fun 6523  df-fn 6524

This theorem is referenced by:  smflimsuplem2  47386