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	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
	fvun1d.2	⊢ ( 𝜑 → 𝐺 Fn 𝐵 )
	fvun1d.3	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) = ∅ )
	fvun1d.4	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
Assertion	fvun1d	⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐺 ) ‘ 𝑋 ) = ( 𝐹 ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		fvun1d.1	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
2		fvun1d.2	⊢ ( 𝜑 → 𝐺 Fn 𝐵 )
3		fvun1d.3	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) = ∅ )
4		fvun1d.4	⊢ ( 𝜑 → 𝑋 ∈ 𝐴 )
5	3 4	jca	⊢ ( 𝜑 → ( ( 𝐴 ∩ 𝐵 ) = ∅ ∧ 𝑋 ∈ 𝐴 ) )
6	1 2 5	3jca	⊢ ( 𝜑 → ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) = ∅ ∧ 𝑋 ∈ 𝐴 ) ) )
7		fvun1	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) = ∅ ∧ 𝑋 ∈ 𝐴 ) ) → ( ( 𝐹 ∪ 𝐺 ) ‘ 𝑋 ) = ( 𝐹 ‘ 𝑋 ) )
8	6 7	syl	⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐺 ) ‘ 𝑋 ) = ( 𝐹 ‘ 𝑋 ) )