Theorem wdomd

Description: Deduce weak dominance from an implicit onto function. (Contributed by Stefan O'Rear, 13-Feb-2015)

	Ref	Expression
Hypotheses	wdomd.b	$$⊢\phi \to B\in W$$
	wdomd.o	$$⊢\left(\phi \wedge x\in A\right)\to \exists y\in Bx=X$$
Assertion	wdomd	$$⊢\phi \to A{\preccurlyeq }^{*}B$$

Proof

Step	Hyp	Ref	Expression
1		wdomd.b	$$⊢\phi \to B\in W$$
2		wdomd.o	$$⊢\left(\phi \wedge x\in A\right)\to \exists y\in Bx=X$$
3		abrexexg	$$⊢B\in W\to \left\{x|\exists y\in Bx=X\right\}\in \mathrm{V}$$
4	1 3	syl	$$⊢\phi \to \left\{x|\exists y\in Bx=X\right\}\in \mathrm{V}$$
5	2	ex	$$⊢\phi \to \left(x\in A\to \exists y\in Bx=X\right)$$
6	5	alrimiv	$$⊢\phi \to \forall x\left(x\in A\to \exists y\in Bx=X\right)$$
7		ssab	$$⊢A\subseteq \left\{x|\exists y\in Bx=X\right\}\leftrightarrow \forall x\left(x\in A\to \exists y\in Bx=X\right)$$
8	6 7	sylibr	$$⊢\phi \to A\subseteq \left\{x|\exists y\in Bx=X\right\}$$
9	4 8	ssexd	$$⊢\phi \to A\in \mathrm{V}$$
10	9 1 2	wdom2d	$$⊢\phi \to A{\preccurlyeq }^{*}B$$