Theorem fnmptif 45372

Description: Functionality and domain of an ordered-pair class abstraction. (Contributed by Glauco Siliprandi, 21-Dec-2024.)

Hypotheses

Ref	Expression
fnmptif.1	⊢ Ⅎ𝑥𝐴
fnmptif.2	⊢ 𝐵 ∈ V
fnmptif.3	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
fnmptif	⊢ 𝐹 Fn 𝐴

Proof of Theorem fnmptif

Step	Hyp	Ref	Expression
1		fnmptif.2	. . . 4 ⊢ 𝐵 ∈ V
2	1	rgenw 3051	. . 3 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V
3		fnmptif.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	3	mptfnf 6616	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
5	2, 4	mpbi 230	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴
6		fnmptif.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	fneq1i 6578	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
8	5, 7	mpbir 231	1 ⊢ 𝐹 Fn 𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2111  Ⅎwnfc 2879  ∀wral 3047  Vcvv 3436   ↦ 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:  dmmptif  45373