Theorem iundif1

Description: Indexed union of class difference with the subtrahend held constant. (Contributed by Brendan Leahy, 6-Aug-2018)

	Ref	Expression
Assertion	iundif1	⊢ ∪ 𝑥 ∈ 𝐴 ( 𝐵 ∖ 𝐶 ) = ( ∪ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		r19.41v	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶 ) )
2		eldif	⊢ ( 𝑦 ∈ ( 𝐵 ∖ 𝐶 ) ↔ ( 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶 ) )
3	2	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∖ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶 ) )
4		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 )
5	4	anbi1i	⊢ ( ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ¬ 𝑦 ∈ 𝐶 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐶 ) )
6	1 3 5	3bitr4i	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∖ 𝐶 ) ↔ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ¬ 𝑦 ∈ 𝐶 ) )
7		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ( 𝐵 ∖ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∖ 𝐶 ) )
8		eldif	⊢ ( 𝑦 ∈ ( ∪ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶 ) ↔ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ¬ 𝑦 ∈ 𝐶 ) )
9	6 7 8	3bitr4i	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ( 𝐵 ∖ 𝐶 ) ↔ 𝑦 ∈ ( ∪ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶 ) )
10	9	eqriv	⊢ ∪ 𝑥 ∈ 𝐴 ( 𝐵 ∖ 𝐶 ) = ( ∪ 𝑥 ∈ 𝐴 𝐵 ∖ 𝐶 )