Theorem fvun2d

Description: The value of a union when the argument is in the second domain, a deduction version. (Contributed by metakunt, 28-May-2024)

	Ref	Expression
Hypotheses	fvun2d.1	$$⊢\phi \to FFnA$$
	fvun2d.2	$$⊢\phi \to GFnB$$
	fvun2d.3	$$⊢\phi \to A\cap B=\varnothing$$
	fvun2d.4	$$⊢\phi \to X\in B$$
Assertion	fvun2d	$$⊢\phi \to \left(F\cup G\right)\left(X\right)=G\left(X\right)$$

Proof

Step	Hyp	Ref	Expression
1		fvun2d.1	$$⊢\phi \to FFnA$$
2		fvun2d.2	$$⊢\phi \to GFnB$$
3		fvun2d.3	$$⊢\phi \to A\cap B=\varnothing$$
4		fvun2d.4	$$⊢\phi \to X\in B$$
5	3 4	jca	$$⊢\phi \to \left(A\cap B=\varnothing \wedge X\in B\right)$$
6	1 2 5	3jca	$$⊢\phi \to \left(FFnA\wedge GFnB\wedge \left(A\cap B=\varnothing \wedge X\in B\right)\right)$$
7		fvun2	$$⊢\left(FFnA\wedge GFnB\wedge \left(A\cap B=\varnothing \wedge X\in B\right)\right)\to \left(F\cup G\right)\left(X\right)=G\left(X\right)$$
8	6 7	syl	$$⊢\phi \to \left(F\cup G\right)\left(X\right)=G\left(X\right)$$