Theorem bnj1386

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1386.1	$$⊢\phi \leftrightarrow \forall f\in A\mathrm{Fun}f$$
	bnj1386.2	$$⊢D=\mathrm{dom}f\cap \mathrm{dom}g$$
	bnj1386.3	$$⊢\psi \leftrightarrow \left(\phi \wedge \forall f\in A\forall g\in A{f\upharpoonright }_D={g\upharpoonright }_D\right)$$
	bnj1386.4	$$⊢x\in A\to \forall fx\in A$$
Assertion	bnj1386	$$⊢\psi \to \mathrm{Fun}\bigcup A$$

Proof

Step	Hyp	Ref	Expression
1		bnj1386.1	$$⊢\phi \leftrightarrow \forall f\in A\mathrm{Fun}f$$
2		bnj1386.2	$$⊢D=\mathrm{dom}f\cap \mathrm{dom}g$$
3		bnj1386.3	$$⊢\psi \leftrightarrow \left(\phi \wedge \forall f\in A\forall g\in A{f\upharpoonright }_D={g\upharpoonright }_D\right)$$
4		bnj1386.4	$$⊢x\in A\to \forall fx\in A$$
5		biid	$$⊢\forall h\in A\mathrm{Fun}h\leftrightarrow \forall h\in A\mathrm{Fun}h$$
6		eqid	$$⊢\mathrm{dom}h\cap \mathrm{dom}g=\mathrm{dom}h\cap \mathrm{dom}g$$
7		biid	$$⊢\left(\forall h\in A\mathrm{Fun}h\wedge \forall h\in A\forall g\in A{h\upharpoonright }_{\left(\mathrm{dom}h\cap \mathrm{dom}g\right)}={g\upharpoonright }_{\left(\mathrm{dom}h\cap \mathrm{dom}g\right)}\right)\leftrightarrow \left(\forall h\in A\mathrm{Fun}h\wedge \forall h\in A\forall g\in A{h\upharpoonright }_{\left(\mathrm{dom}h\cap \mathrm{dom}g\right)}={g\upharpoonright }_{\left(\mathrm{dom}h\cap \mathrm{dom}g\right)}\right)$$
8	1 2 3 4 5 6 7	bnj1385	$$⊢\psi \to \mathrm{Fun}\bigcup A$$