Theorem fvun1d

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

	Ref	Expression
Hypotheses	fvun1d.1	$$⊢\phi \to FFnA$$
	fvun1d.2	$$⊢\phi \to GFnB$$
	fvun1d.3	$$⊢\phi \to A\cap B=\varnothing$$
	fvun1d.4	$$⊢\phi \to X\in A$$
Assertion	fvun1d	$$⊢\phi \to \left(F\cup G\right)\left(X\right)=F\left(X\right)$$

Proof

Step	Hyp	Ref	Expression
1		fvun1d.1	$$⊢\phi \to FFnA$$
2		fvun1d.2	$$⊢\phi \to GFnB$$
3		fvun1d.3	$$⊢\phi \to A\cap B=\varnothing$$
4		fvun1d.4	$$⊢\phi \to X\in A$$
5	3 4	jca	$$⊢\phi \to \left(A\cap B=\varnothing \wedge X\in A\right)$$
6	1 2 5	3jca	$$⊢\phi \to \left(FFnA\wedge GFnB\wedge \left(A\cap B=\varnothing \wedge X\in A\right)\right)$$
7		fvun1	$$⊢\left(FFnA\wedge GFnB\wedge \left(A\cap B=\varnothing \wedge X\in A\right)\right)\to \left(F\cup G\right)\left(X\right)=F\left(X\right)$$
8	6 7	syl	$$⊢\phi \to \left(F\cup G\right)\left(X\right)=F\left(X\right)$$