Theorem fnmptd 6641

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 412	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉))
4	1, 3	ralrimi 3236	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
5		fnmptd.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fnmpt 6640	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴)
7	4, 6	syl 17	1 ⊢ (𝜑 → 𝐹 Fn 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3052   ↦ cmpt 5181   Fn wfn 6495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-fun 6502  df-fn 6503

This theorem is referenced by:  rngqiprngimf1  21270  suppgsumssiun  33170  elrgspnsubrunlem2  33346  nsgmgc  33509  nsgqusf1o  33513  elrspunsn  33526  ply1gsumz  33696  psrgsum  33729  psrmonprod  33733  esplyfvaln  33755  esplyind  33756  ply1degltdimlem  33804  evls1fldgencl  33852  extdgfialglem2  33875  ply1annidllem  33883  algextdeglem6  33904  limsupequzmptlem  46090  liminfval2  46130  smflimmpt  47172  smflimsuplem7  47188  cfsetsnfsetfo  47424