Theorem dmmptd

Description: The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019)

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

Proof

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