Theorem fnmptd 6626

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 413	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉))
4	1, 3	ralrimi 3237	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
5		fnmptd.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fnmpt 6625	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴)
7	4, 6	syl 17	1 ⊢ (𝜑 → 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  Ⅎwnf 1790   ∈ wcel 2119  ∀wral 3053   ↦ cmpt 5153   Fn wfn 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-fun 6487  df-fn 6488

This theorem is referenced by:  rngqiprngimf1  21293  suppgsumssiun  33153  elrgspnsubrunlem2  33329  nsgmgc  33495  nsgqusf1o  33499  elrspunsn  33512  ply1gsumz  33682  psrgsum  33732  psrmonprod  33736  esplyfvaln  33758  esplyind  33759  ply1degltdimlem  33806  evls1fldgencl  33854  extdgfialglem2  33877  ply1annidllem  33885  algextdeglem6  33906  limsupequzmptlem  46171  liminfval2  46211  smflimmpt  47253  smflimsuplem7  47269  cfsetsnfsetfo  47523