Theorem fnmptif 46416

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 3083	. . 3 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V
3		fnmptif.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4	3	mptfnf 6682	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
5	2, 4	mpbi 233	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴
6		fnmptif.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
7	6	fneq1i 6644	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
8	5, 7	mpbir 234	1 ⊢ 𝐹 Fn 𝐴

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2146  Ⅎwnfc 2910  ∀wral 3079  Vcvv 3453   ↦ cmpt 5188   Fn wfn 6541

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-fun 6548  df-fn 6549

This theorem is used by:  dmmptif  46417