Theorem fnmptif 44671

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 3062	. . 3 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V
3		fnmptif.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	3	mptfnf 6695	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
5	2, 4	mpbi 229	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴
6		fnmptif.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	fneq1i 6656	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
8	5, 7	mpbir 230	1 ⊢ 𝐹 Fn 𝐴

Syntax hints:   = wceq 1533   ∈ wcel 2098  Ⅎwnfc 2879  ∀wral 3058  Vcvv 3473   ↦ cmpt 5235   Fn wfn 6548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-fun 6555  df-fn 6556

This theorem is referenced by:  dmmptif  44672