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Metamath Proof Explorer

Theorem brcnvrabga

Description: The law of concretion for the converse of operation class abstraction. (Contributed by Peter Mazsa, 25-Oct-2022)

Ref Expression
Hypotheses brrabga.1 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝜑 ↔ 𝜓 ) )
brcnvrabga.2 ⊢ 𝑅 = ◡ { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ 𝜑 }
Assertion brcnvrabga ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 𝑅 ⟨ 𝐵 , 𝐶 ⟩ ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 brrabga.1 ⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ) → ( 𝜑 ↔ 𝜓 ) )
2 brcnvrabga.2 ⊢ 𝑅 = ◡ { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ 𝜑 }
3 relcnv ⊢ Rel ◡ { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ 𝜑 }
4 2 releqi ⊢ ( Rel 𝑅 ↔ Rel ◡ { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ 𝜑 } )
5 3 4 mpbir ⊢ Rel 𝑅
6 5 relbrcnv ⊢ ( ⟨ 𝐵 , 𝐶 ⟩ ◡ 𝑅 𝐴 ↔ 𝐴 𝑅 ⟨ 𝐵 , 𝐶 ⟩ )
7 1 3coml ⊢ ( ( 𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝑥 = 𝐴 ) → ( 𝜑 ↔ 𝜓 ) )
8 2 cnveqi ⊢ ◡ 𝑅 = ◡ ◡ { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ 𝜑 }
9 reloprab ⊢ Rel { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ 𝜑 }
10 dfrel2 ⊢ ( Rel { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ 𝜑 } ↔ ◡ ◡ { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ 𝜑 } = { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ 𝜑 } )
11 9 10 mpbi ⊢ ◡ ◡ { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ 𝜑 } = { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ 𝜑 }
12 8 11 eqtri ⊢ ◡ 𝑅 = { ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑥 ⟩ ∣ 𝜑 }
13 7 12 brrabga ⊢ ( ( 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉 ) → ( ⟨ 𝐵 , 𝐶 ⟩ ◡ 𝑅 𝐴 ↔ 𝜓 ) )
14 13 3comr ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ⟨ 𝐵 , 𝐶 ⟩ ◡ 𝑅 𝐴 ↔ 𝜓 ) )
15 6 14 bitr3id ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( 𝐴 𝑅 ⟨ 𝐵 , 𝐶 ⟩ ↔ 𝜓 ) )