Theorem abfmpunirn

Description: Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016)

	Ref	Expression
Hypotheses	abfmpunirn.1	⊢ 𝐹 = ( 𝑥 ∈ 𝑉 ↦ { 𝑦 ∣ 𝜑 } )
	abfmpunirn.2	⊢ { 𝑦 ∣ 𝜑 } ∈ V
	abfmpunirn.3	⊢ ( 𝑦 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
Assertion	abfmpunirn	⊢ ( 𝐵 ∈ ∪ ran 𝐹 ↔ ( 𝐵 ∈ V ∧ ∃ 𝑥 ∈ 𝑉 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		abfmpunirn.1	⊢ 𝐹 = ( 𝑥 ∈ 𝑉 ↦ { 𝑦 ∣ 𝜑 } )
2		abfmpunirn.2	⊢ { 𝑦 ∣ 𝜑 } ∈ V
3		abfmpunirn.3	⊢ ( 𝑦 = 𝐵 → ( 𝜑 ↔ 𝜓 ) )
4		elex	⊢ ( 𝐵 ∈ ∪ ran 𝐹 → 𝐵 ∈ V )
5	2 1	fnmpti	⊢ 𝐹 Fn 𝑉
6		fnunirn	⊢ ( 𝐹 Fn 𝑉 → ( 𝐵 ∈ ∪ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝑉 𝐵 ∈ ( 𝐹 ‘ 𝑥 ) ) )
7	5 6	ax-mp	⊢ ( 𝐵 ∈ ∪ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝑉 𝐵 ∈ ( 𝐹 ‘ 𝑥 ) )
8	1	fvmpt2	⊢ ( ( 𝑥 ∈ 𝑉 ∧ { 𝑦 ∣ 𝜑 } ∈ V ) → ( 𝐹 ‘ 𝑥 ) = { 𝑦 ∣ 𝜑 } )
9	2 8	mpan2	⊢ ( 𝑥 ∈ 𝑉 → ( 𝐹 ‘ 𝑥 ) = { 𝑦 ∣ 𝜑 } )
10	9	eleq2d	⊢ ( 𝑥 ∈ 𝑉 → ( 𝐵 ∈ ( 𝐹 ‘ 𝑥 ) ↔ 𝐵 ∈ { 𝑦 ∣ 𝜑 } ) )
11	10	rexbiia	⊢ ( ∃ 𝑥 ∈ 𝑉 𝐵 ∈ ( 𝐹 ‘ 𝑥 ) ↔ ∃ 𝑥 ∈ 𝑉 𝐵 ∈ { 𝑦 ∣ 𝜑 } )
12	7 11	bitri	⊢ ( 𝐵 ∈ ∪ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝑉 𝐵 ∈ { 𝑦 ∣ 𝜑 } )
13	3	elabg	⊢ ( 𝐵 ∈ V → ( 𝐵 ∈ { 𝑦 ∣ 𝜑 } ↔ 𝜓 ) )
14	13	rexbidv	⊢ ( 𝐵 ∈ V → ( ∃ 𝑥 ∈ 𝑉 𝐵 ∈ { 𝑦 ∣ 𝜑 } ↔ ∃ 𝑥 ∈ 𝑉 𝜓 ) )
15	12 14	syl5bb	⊢ ( 𝐵 ∈ V → ( 𝐵 ∈ ∪ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝑉 𝜓 ) )
16	4 15	biadanii	⊢ ( 𝐵 ∈ ∪ ran 𝐹 ↔ ( 𝐵 ∈ V ∧ ∃ 𝑥 ∈ 𝑉 𝜓 ) )