Theorem cic1st2ndbr

Description: Rewrite the predicate of isomorphic objects with separated parts. (Contributed by Zhi Wang, 27-Oct-2025)

	Ref	Expression
Assertion	cic1st2ndbr	⊢ ( 𝑃 ∈ ( ≃𝑐 ‘ 𝐶 ) → ( 1st ‘ 𝑃 ) ( ≃𝑐 ‘ 𝐶 ) ( 2nd ‘ 𝑃 ) )

Proof

Step	Hyp	Ref	Expression
1		cic1st2nd	⊢ ( 𝑃 ∈ ( ≃𝑐 ‘ 𝐶 ) → 𝑃 = ⟨ ( 1st ‘ 𝑃 ) , ( 2nd ‘ 𝑃 ) ⟩ )
2		id	⊢ ( 𝑃 ∈ ( ≃𝑐 ‘ 𝐶 ) → 𝑃 ∈ ( ≃𝑐 ‘ 𝐶 ) )
3	1 2	eqeltrrd	⊢ ( 𝑃 ∈ ( ≃𝑐 ‘ 𝐶 ) → ⟨ ( 1st ‘ 𝑃 ) , ( 2nd ‘ 𝑃 ) ⟩ ∈ ( ≃𝑐 ‘ 𝐶 ) )
4		df-br	⊢ ( ( 1st ‘ 𝑃 ) ( ≃𝑐 ‘ 𝐶 ) ( 2nd ‘ 𝑃 ) ↔ ⟨ ( 1st ‘ 𝑃 ) , ( 2nd ‘ 𝑃 ) ⟩ ∈ ( ≃𝑐 ‘ 𝐶 ) )
5	3 4	sylibr	⊢ ( 𝑃 ∈ ( ≃𝑐 ‘ 𝐶 ) → ( 1st ‘ 𝑃 ) ( ≃𝑐 ‘ 𝐶 ) ( 2nd ‘ 𝑃 ) )