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	$$⊢\phi \to \left(x\in A⟼B\right)\in MblFn$$
	mbfmptcl.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
Assertion	mbfmptcl	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℂ$$

Proof

Step	Hyp	Ref	Expression
1		mbfmptcl.1	$$⊢\phi \to \left(x\in A⟼B\right)\in MblFn$$
2		mbfmptcl.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
3		mbff	$$⊢\left(x\in A⟼B\right)\in MblFn\to \left(x\in A⟼B\right):\mathrm{dom}\left(x\in A⟼B\right)⟶ℂ$$
4	1 3	syl	$$⊢\phi \to \left(x\in A⟼B\right):\mathrm{dom}\left(x\in A⟼B\right)⟶ℂ$$
5	2	ralrimiva	$$⊢\phi \to \forall x\in AB\in V$$
6		dmmptg	$$⊢\forall x\in AB\in V\to \mathrm{dom}\left(x\in A⟼B\right)=A$$
7	5 6	syl	$$⊢\phi \to \mathrm{dom}\left(x\in A⟼B\right)=A$$
8	7	feq2d	$$⊢\phi \to \left(\left(x\in A⟼B\right):\mathrm{dom}\left(x\in A⟼B\right)⟶ℂ\leftrightarrow \left(x\in A⟼B\right):A⟶ℂ\right)$$
9	4 8	mpbid	$$⊢\phi \to \left(x\in A⟼B\right):A⟶ℂ$$
10	9	fvmptelrn	$$⊢\left(\phi \wedge x\in A\right)\to B\in ℂ$$