Theorem nat1st2nd

Description: Rewrite the natural transformation predicate with separated functor parts. (Contributed by Mario Carneiro, 6-Jan-2017)

	Ref	Expression
Hypotheses	natrcl.1	⊢ 𝑁 = ( 𝐶 Nat 𝐷 )
	nat1st2nd.2	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐹 𝑁 𝐺 ) )
Assertion	nat1st2nd	⊢ ( 𝜑 → 𝐴 ∈ ( ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ 𝑁 ⟨ ( 1st ‘ 𝐺 ) , ( 2nd ‘ 𝐺 ) ⟩ ) )

Proof

Step	Hyp	Ref	Expression
1		natrcl.1	⊢ 𝑁 = ( 𝐶 Nat 𝐷 )
2		nat1st2nd.2	⊢ ( 𝜑 → 𝐴 ∈ ( 𝐹 𝑁 𝐺 ) )
3		relfunc	⊢ Rel ( 𝐶 Func 𝐷 )
4	1	natrcl	⊢ ( 𝐴 ∈ ( 𝐹 𝑁 𝐺 ) → ( 𝐹 ∈ ( 𝐶 Func 𝐷 ) ∧ 𝐺 ∈ ( 𝐶 Func 𝐷 ) ) )
5	2 4	syl	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝐶 Func 𝐷 ) ∧ 𝐺 ∈ ( 𝐶 Func 𝐷 ) ) )
6	5	simpld	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 Func 𝐷 ) )
7		1st2nd	⊢ ( ( Rel ( 𝐶 Func 𝐷 ) ∧ 𝐹 ∈ ( 𝐶 Func 𝐷 ) ) → 𝐹 = ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ )
8	3 6 7	sylancr	⊢ ( 𝜑 → 𝐹 = ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ )
9	5	simprd	⊢ ( 𝜑 → 𝐺 ∈ ( 𝐶 Func 𝐷 ) )
10		1st2nd	⊢ ( ( Rel ( 𝐶 Func 𝐷 ) ∧ 𝐺 ∈ ( 𝐶 Func 𝐷 ) ) → 𝐺 = ⟨ ( 1st ‘ 𝐺 ) , ( 2nd ‘ 𝐺 ) ⟩ )
11	3 9 10	sylancr	⊢ ( 𝜑 → 𝐺 = ⟨ ( 1st ‘ 𝐺 ) , ( 2nd ‘ 𝐺 ) ⟩ )
12	8 11	oveq12d	⊢ ( 𝜑 → ( 𝐹 𝑁 𝐺 ) = ( ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ 𝑁 ⟨ ( 1st ‘ 𝐺 ) , ( 2nd ‘ 𝐺 ) ⟩ ) )
13	2 12	eleqtrd	⊢ ( 𝜑 → 𝐴 ∈ ( ⟨ ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) ⟩ 𝑁 ⟨ ( 1st ‘ 𝐺 ) , ( 2nd ‘ 𝐺 ) ⟩ ) )