Theorem dmmpt1 45844

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 2763	. 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)
4		dmmpt1.c	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)
5	1, 2, 3, 4	dmmptdff 45800	1 ⊢ (𝜑 → dom (𝑥 ∈ 𝐵 ↦ 𝐶) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561  Ⅎwnf 1804   ∈ wcel 2143  Ⅎwnfc 2910   ↦ cmpt 5182  dom cdm 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3078  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-mpt 5183  df-xp 5654  df-rel 5655  df-cnv 5656  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661

This theorem is referenced by:  smffmptf  47379