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Metamath Proof Explorer

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 elecreseq ⊢ ( 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 ) ) )