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Metamath Proof Explorer

Theorem brabgaf

Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013) (Revised by Thierry Arnoux, 17-May-2020)

Ref Expression
Hypotheses brabgaf.0 ⊢ Ⅎ 𝑥 𝜓
brabgaf.1 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
brabgaf.2 ⊢ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
Assertion brabgaf ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 𝑅 𝐵 ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 brabgaf.0 ⊢ Ⅎ 𝑥 𝜓
2 brabgaf.1 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ 𝜓 ) )
3 brabgaf.2 ⊢ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 }
4 df-br ⊢ ( 𝐴 𝑅 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 )
5 3 eleq2i ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ 𝑅 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } )
6 4 5 bitri ⊢ ( 𝐴 𝑅 𝐵 ↔ ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } )
7 elopab ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) )
8 elisset ⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
9 elisset ⊢ ( 𝐵 ∈ 𝑊 → ∃ 𝑦 𝑦 = 𝐵 )
10 exdistrv ⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) )
11 nfe1 ⊢ Ⅎ 𝑥 ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 )
12 11 1 nfbi ⊢ Ⅎ 𝑥 ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 )
13 nfe1 ⊢ Ⅎ 𝑦 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 )
14 13 nfex ⊢ Ⅎ 𝑦 ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 )
15 nfv ⊢ Ⅎ 𝑦 𝜓
16 14 15 nfbi ⊢ Ⅎ 𝑦 ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 )
17 opeq12 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ )
18 copsexgw ⊢ ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
19 18 eqcoms ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ = ⟨ 𝐴 , 𝐵 ⟩ → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
20 17 19 syl ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜑 ↔ ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ) )
21 20 2 bitr3d ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )
22 16 21 exlimi ⊢ ( ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )
23 12 22 exlimi ⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )
24 10 23 sylbir ⊢ ( ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃ 𝑦 𝑦 = 𝐵 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )
25 8 9 24 syl2an ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 ∃ 𝑦 ( ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝑥 , 𝑦 ⟩ ∧ 𝜑 ) ↔ 𝜓 ) )
26 7 25 bitrid ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ⟨ 𝐴 , 𝐵 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝜑 } ↔ 𝜓 ) )
27 6 26 bitrid ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 𝑅 𝐵 ↔ 𝜓 ) )