Theorem mbfmptcl

Description: Lemma for the MblFn predicate applied to a mapping operation. (Contributed by Mario Carneiro, 11-Aug-2014)

	Ref	Expression
Hypotheses	mbfmptcl.1	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn )
	mbfmptcl.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
Assertion	mbfmptcl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		mbfmptcl.1	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn )
2		mbfmptcl.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
3		mbff	⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⟶ ℂ )
4	1 3	syl	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⟶ ℂ )
5	2	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 )
6		dmmptg	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 )
7	5 6	syl	⊢ ( 𝜑 → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 )
8	7	feq2d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⟶ ℂ ↔ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ) )
9	4 8	mpbid	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ )
10	9	fvmptelrn	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ )