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 ) )