Theorem natffn

Description: The natural transformation set operation is a well-defined function. (Contributed by Mario Carneiro, 12-Jan-2017)

	Ref	Expression
Hypothesis	natrcl.1	⊢ 𝑁 = ( 𝐶 Nat 𝐷 )
Assertion	natffn	⊢ 𝑁 Fn ( ( 𝐶 Func 𝐷 ) × ( 𝐶 Func 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		natrcl.1	⊢ 𝑁 = ( 𝐶 Nat 𝐷 )
2		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
3		eqid	⊢ ( Hom ‘ 𝐶 ) = ( Hom ‘ 𝐶 )
4		eqid	⊢ ( Hom ‘ 𝐷 ) = ( Hom ‘ 𝐷 )
5		eqid	⊢ ( comp ‘ 𝐷 ) = ( comp ‘ 𝐷 )
6	1 2 3 4 5	natfval	⊢ 𝑁 = ( 𝑓 ∈ ( 𝐶 Func 𝐷 ) , 𝑔 ∈ ( 𝐶 Func 𝐷 ) ↦ ⦋ ( 1st ‘ 𝑓 ) / 𝑟 ⦌ ⦋ ( 1st ‘ 𝑔 ) / 𝑠 ⦌ { 𝑎 ∈ X 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( 𝑟 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝑠 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( ⟨ ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) ⟩ ( comp ‘ 𝐷 ) ( 𝑠 ‘ 𝑦 ) ) ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) ( ⟨ ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) ⟩ ( comp ‘ 𝐷 ) ( 𝑠 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } )
7		ovex	⊢ ( ( 𝑟 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝑠 ‘ 𝑥 ) ) ∈ V
8	7	rgenw	⊢ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( 𝑟 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝑠 ‘ 𝑥 ) ) ∈ V
9		ixpexg	⊢ ( ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( 𝑟 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝑠 ‘ 𝑥 ) ) ∈ V → X 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( 𝑟 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝑠 ‘ 𝑥 ) ) ∈ V )
10	8 9	ax-mp	⊢ X 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( 𝑟 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝑠 ‘ 𝑥 ) ) ∈ V
11	10	rabex	⊢ { 𝑎 ∈ X 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( 𝑟 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝑠 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( ⟨ ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) ⟩ ( comp ‘ 𝐷 ) ( 𝑠 ‘ 𝑦 ) ) ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) ( ⟨ ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) ⟩ ( comp ‘ 𝐷 ) ( 𝑠 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } ∈ V
12	11	csbex	⊢ ⦋ ( 1st ‘ 𝑔 ) / 𝑠 ⦌ { 𝑎 ∈ X 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( 𝑟 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝑠 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( ⟨ ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) ⟩ ( comp ‘ 𝐷 ) ( 𝑠 ‘ 𝑦 ) ) ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) ( ⟨ ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) ⟩ ( comp ‘ 𝐷 ) ( 𝑠 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } ∈ V
13	12	csbex	⊢ ⦋ ( 1st ‘ 𝑓 ) / 𝑟 ⦌ ⦋ ( 1st ‘ 𝑔 ) / 𝑠 ⦌ { 𝑎 ∈ X 𝑥 ∈ ( Base ‘ 𝐶 ) ( ( 𝑟 ‘ 𝑥 ) ( Hom ‘ 𝐷 ) ( 𝑠 ‘ 𝑥 ) ) ∣ ∀ 𝑥 ∈ ( Base ‘ 𝐶 ) ∀ 𝑦 ∈ ( Base ‘ 𝐶 ) ∀ ℎ ∈ ( 𝑥 ( Hom ‘ 𝐶 ) 𝑦 ) ( ( 𝑎 ‘ 𝑦 ) ( ⟨ ( 𝑟 ‘ 𝑥 ) , ( 𝑟 ‘ 𝑦 ) ⟩ ( comp ‘ 𝐷 ) ( 𝑠 ‘ 𝑦 ) ) ( ( 𝑥 ( 2nd ‘ 𝑓 ) 𝑦 ) ‘ ℎ ) ) = ( ( ( 𝑥 ( 2nd ‘ 𝑔 ) 𝑦 ) ‘ ℎ ) ( ⟨ ( 𝑟 ‘ 𝑥 ) , ( 𝑠 ‘ 𝑥 ) ⟩ ( comp ‘ 𝐷 ) ( 𝑠 ‘ 𝑦 ) ) ( 𝑎 ‘ 𝑥 ) ) } ∈ V
14	6 13	fnmpoi	⊢ 𝑁 Fn ( ( 𝐶 Func 𝐷 ) × ( 𝐶 Func 𝐷 ) )