Theorem fnmptif 43843

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 3066	. . 3 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V
3		fnmptif.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	3	mptfnf 6675	. . 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 6638	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
8	5, 7	mpbir 230	1 ⊢ 𝐹 Fn 𝐴

Syntax hints:   = wceq 1542   ∈ wcel 2107  Ⅎwnfc 2884  ∀wral 3062  Vcvv 3475   ↦ cmpt 5227   Fn wfn 6530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-fun 6537  df-fn 6538

This theorem is referenced by:  dmmptif  43844