Theorem dmmptdf

Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypotheses	dmmptdf.x	⊢ Ⅎ 𝑥 𝜑
	dmmptdf.a	⊢ 𝐴 = ( 𝑥 ∈ 𝐵 ↦ 𝐶 )
	dmmptdf.c	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 ∈ 𝑉 )
Assertion	dmmptdf	⊢ ( 𝜑 → dom 𝐴 = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		dmmptdf.x	⊢ Ⅎ 𝑥 𝜑
2		dmmptdf.a	⊢ 𝐴 = ( 𝑥 ∈ 𝐵 ↦ 𝐶 )
3		dmmptdf.c	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 ∈ 𝑉 )
4	2	dmmpt	⊢ dom 𝐴 = { 𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V }
5	3	elexd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 ∈ V )
6	1 5	ralrimia	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 𝐶 ∈ V )
7		rabid2	⊢ ( 𝐵 = { 𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V } ↔ ∀ 𝑥 ∈ 𝐵 𝐶 ∈ V )
8	6 7	sylibr	⊢ ( 𝜑 → 𝐵 = { 𝑥 ∈ 𝐵 ∣ 𝐶 ∈ V } )
9	4 8	eqtr4id	⊢ ( 𝜑 → dom 𝐴 = 𝐵 )