Theorem ac6c5

Description: Equivalent of Axiom of Choice. B is a collection B ( x ) of nonempty sets. Remark after Theorem 10.46 of TakeutiZaring p. 98. (Contributed by Mario Carneiro, 22-Mar-2013)

	Ref	Expression
Hypotheses	ac6c4.1	$$⊢A\in \mathrm{V}$$
	ac6c4.2	$$⊢B\in \mathrm{V}$$
Assertion	ac6c5	$$⊢\forall x\in AB\ne \varnothing \to \exists f\forall x\in Af\left(x\right)\in B$$

Proof

Step	Hyp	Ref	Expression
1		ac6c4.1	$$⊢A\in \mathrm{V}$$
2		ac6c4.2	$$⊢B\in \mathrm{V}$$
3	1 2	ac6c4	$$⊢\forall x\in AB\ne \varnothing \to \exists f\left(fFnA\wedge \forall x\in Af\left(x\right)\in B\right)$$
4		exsimpr	$$⊢\exists f\left(fFnA\wedge \forall x\in Af\left(x\right)\in B\right)\to \exists f\forall x\in Af\left(x\right)\in B$$
5	3 4	syl	$$⊢\forall x\in AB\ne \varnothing \to \exists f\forall x\in Af\left(x\right)\in B$$