Theorem eldmqsres

Description: Elementhood in a restricted domain quotient set. (Contributed by Peter Mazsa, 21-Aug-2020)

	Ref	Expression
Assertion	eldmqsres	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ ( dom ( 𝑅 ↾ 𝐴 ) / ( 𝑅 ↾ 𝐴 ) ) ↔ ∃ 𝑢 ∈ 𝐴 ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elqsg	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ ( dom ( 𝑅 ↾ 𝐴 ) / ( 𝑅 ↾ 𝐴 ) ) ↔ ∃ 𝑢 ∈ dom ( 𝑅 ↾ 𝐴 ) 𝐵 = [ 𝑢 ] ( 𝑅 ↾ 𝐴 ) ) )
2		eldmres2	⊢ ( 𝑢 ∈ V → ( 𝑢 ∈ dom ( 𝑅 ↾ 𝐴 ) ↔ ( 𝑢 ∈ 𝐴 ∧ ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ) ) )
3	2	elv	⊢ ( 𝑢 ∈ dom ( 𝑅 ↾ 𝐴 ) ↔ ( 𝑢 ∈ 𝐴 ∧ ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ) )
4	3	anbi1i	⊢ ( ( 𝑢 ∈ dom ( 𝑅 ↾ 𝐴 ) ∧ 𝐵 = [ 𝑢 ] ( 𝑅 ↾ 𝐴 ) ) ↔ ( ( 𝑢 ∈ 𝐴 ∧ ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ) ∧ 𝐵 = [ 𝑢 ] ( 𝑅 ↾ 𝐴 ) ) )
5		ecres2	⊢ ( 𝑢 ∈ 𝐴 → [ 𝑢 ] ( 𝑅 ↾ 𝐴 ) = [ 𝑢 ] 𝑅 )
6	5	eqeq2d	⊢ ( 𝑢 ∈ 𝐴 → ( 𝐵 = [ 𝑢 ] ( 𝑅 ↾ 𝐴 ) ↔ 𝐵 = [ 𝑢 ] 𝑅 ) )
7	6	pm5.32i	⊢ ( ( 𝑢 ∈ 𝐴 ∧ 𝐵 = [ 𝑢 ] ( 𝑅 ↾ 𝐴 ) ) ↔ ( 𝑢 ∈ 𝐴 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) )
8	7	anbi2i	⊢ ( ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ ( 𝑢 ∈ 𝐴 ∧ 𝐵 = [ 𝑢 ] ( 𝑅 ↾ 𝐴 ) ) ) ↔ ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ ( 𝑢 ∈ 𝐴 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) ) )
9		an21	⊢ ( ( ( 𝑢 ∈ 𝐴 ∧ ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ) ∧ 𝐵 = [ 𝑢 ] ( 𝑅 ↾ 𝐴 ) ) ↔ ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ ( 𝑢 ∈ 𝐴 ∧ 𝐵 = [ 𝑢 ] ( 𝑅 ↾ 𝐴 ) ) ) )
10		an12	⊢ ( ( 𝑢 ∈ 𝐴 ∧ ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) ) ↔ ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ ( 𝑢 ∈ 𝐴 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) ) )
11	8 9 10	3bitr4i	⊢ ( ( ( 𝑢 ∈ 𝐴 ∧ ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ) ∧ 𝐵 = [ 𝑢 ] ( 𝑅 ↾ 𝐴 ) ) ↔ ( 𝑢 ∈ 𝐴 ∧ ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) ) )
12	4 11	bitri	⊢ ( ( 𝑢 ∈ dom ( 𝑅 ↾ 𝐴 ) ∧ 𝐵 = [ 𝑢 ] ( 𝑅 ↾ 𝐴 ) ) ↔ ( 𝑢 ∈ 𝐴 ∧ ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) ) )
13	12	rexbii2	⊢ ( ∃ 𝑢 ∈ dom ( 𝑅 ↾ 𝐴 ) 𝐵 = [ 𝑢 ] ( 𝑅 ↾ 𝐴 ) ↔ ∃ 𝑢 ∈ 𝐴 ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) )
14	1 13	bitrdi	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ ( dom ( 𝑅 ↾ 𝐴 ) / ( 𝑅 ↾ 𝐴 ) ) ↔ ∃ 𝑢 ∈ 𝐴 ( ∃ 𝑥 𝑥 ∈ [ 𝑢 ] 𝑅 ∧ 𝐵 = [ 𝑢 ] 𝑅 ) ) )