Theorem fnmptif 46000

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 6670	. . 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 6632	. 2 ⊢ (𝐹 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
8	5, 7	mpbir 234	1 ⊢ 𝐹 Fn 𝐴

Syntax hints:   = wceq 1570   ∈ wcel 2143  Ⅎwnfc 2910  ∀wral 3079  Vcvv 3455   ↦ cmpt 5192   Fn wfn 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  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 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-fun 6538  df-fn 6539

This theorem is referenced by:  dmmptif  46001