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	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
	fvun2d.2	⊢ ( 𝜑 → 𝐺 Fn 𝐵 )
	fvun2d.3	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) = ∅ )
	fvun2d.4	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	fvun2d	⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐺 ) ‘ 𝑋 ) = ( 𝐺 ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		fvun2d.1	⊢ ( 𝜑 → 𝐹 Fn 𝐴 )
2		fvun2d.2	⊢ ( 𝜑 → 𝐺 Fn 𝐵 )
3		fvun2d.3	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) = ∅ )
4		fvun2d.4	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5	3 4	jca	⊢ ( 𝜑 → ( ( 𝐴 ∩ 𝐵 ) = ∅ ∧ 𝑋 ∈ 𝐵 ) )
6	1 2 5	3jca	⊢ ( 𝜑 → ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) = ∅ ∧ 𝑋 ∈ 𝐵 ) ) )
7		fvun2	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ( ( 𝐴 ∩ 𝐵 ) = ∅ ∧ 𝑋 ∈ 𝐵 ) ) → ( ( 𝐹 ∪ 𝐺 ) ‘ 𝑋 ) = ( 𝐺 ‘ 𝑋 ) )
8	6 7	syl	⊢ ( 𝜑 → ( ( 𝐹 ∪ 𝐺 ) ‘ 𝑋 ) = ( 𝐺 ‘ 𝑋 ) )