Theorem mpoexd

Description: Existence of an operation class abstraction. (Contributed by Thierry Arnoux, 17-Jun-2026)

	Ref	Expression
Hypotheses	mpoexd.1	$$⊢\phi \to A\in V$$
	mpoexd.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in W$$
Assertion	mpoexd	$$⊢\phi \to \left(x\in A,y\in B⟼C\right)\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		mpoexd.1	$$⊢\phi \to A\in V$$
2		mpoexd.2	$$⊢\left(\phi \wedge x\in A\right)\to B\in W$$
3	2	ralrimiva	$$⊢\phi \to \forall x\in AB\in W$$
4		eqid	$$⊢\left(x\in A,y\in B⟼C\right)=\left(x\in A,y\in B⟼C\right)$$
5	4	mpoexxg	$$⊢\left(A\in V\wedge \forall x\in AB\in W\right)\to \left(x\in A,y\in B⟼C\right)\in \mathrm{V}$$
6	1 3 5	syl2anc	$$⊢\phi \to \left(x\in A,y\in B⟼C\right)\in \mathrm{V}$$