Theorem dmmpt1 45272

Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 5-Jan-2025.)

Hypotheses

Ref	Expression
dmmpt1.x	⊢ Ⅎ𝑥𝜑
dmmpt1.1	⊢ Ⅎ𝑥𝐵
dmmpt1.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
dmmpt1	⊢ (𝜑 → dom (𝑥 ∈ 𝐵 ↦ 𝐶) = 𝐵)

Proof of Theorem dmmpt1

Step	Hyp	Ref	Expression
1		dmmpt1.x	. 2 ⊢ Ⅎ𝑥𝜑
2		dmmpt1.1	. 2 ⊢ Ⅎ𝑥𝐵
3		eqid 2736	. 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)
4		dmmpt1.c	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
5	1, 2, 3, 4	dmmptdff 45227	1 ⊢ (𝜑 → dom (𝑥 ∈ 𝐵 ↦ 𝐶) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2109  Ⅎwnfc 2884   ↦ cmpt 5206  dom cdm 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-mpt 5207  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672

This theorem is referenced by:  smffmptf  46813