Theorem fnmptd 6662

Description: The maps-to notation defines a function with domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
fnmptd.1	⊢ Ⅎ𝑥𝜑
fnmptd.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
fnmptd.3	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
fnmptd	⊢ (𝜑 → 𝐹 Fn 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fnmptd

Step	Hyp	Ref	Expression
1		fnmptd.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		fnmptd.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
3	2	ex 416	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉))
4	1, 3	ralrimi 3260	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
5		fnmptd.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fnmpt 6661	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴)
7	4, 6	syl 17	1 ⊢ (𝜑 → 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560  Ⅎwnf 1803   ∈ wcel 2142  ∀wral 3076   ↦ cmpt 5181   Fn wfn 6516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-fun 6523  df-fn 6524

This theorem is referenced by:  rngqiprngimf1  21370  suppgsumssiun  33252  elrgspnsubrunlem2  33429  nsgmgc  33598  nsgqusf1o  33602  elrspunsn  33615  ply1gsumz  33795  psrgsum  33845  psrmonprod  33849  esplyfvaln  33871  esplyind  33872  ply1degltdimlem  33919  evls1fldgencl  33967  extdgfialglem2  33990  ply1annidllem  33998  algextdeglem6  34019  limsupequzmptlem  46302  liminfval2  46342  smflimmpt  47384  smflimsuplem7  47400  cfsetsnfsetfo  47654