Theorem fnmptd 6622

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 3230	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉)
5		fnmptd.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
6	5	fnmpt 6621	. 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 1541  Ⅎwnf 1784   ∈ wcel 2111  ∀wral 3047   ↦ cmpt 5170   Fn wfn 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-fun 6483  df-fn 6484

This theorem is referenced by:  rngqiprngimf1  21237  elrgspnsubrunlem2  33215  nsgmgc  33377  nsgqusf1o  33381  elrspunsn  33394  ply1gsumz  33559  ply1degltdimlem  33635  evls1fldgencl  33683  extdgfialglem2  33706  ply1annidllem  33714  algextdeglem6  33735  limsupequzmptlem  45836  liminfval2  45876  smflimmpt  46918  smflimsuplem7  46934  cfsetsnfsetfo  47170