Theorem natffn 18026

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

Hypothesis

Ref	Expression
natrcl.1	⊢ 𝑁 = (𝐶 Nat 𝐷)

Assertion

Ref	Expression
natffn	⊢ 𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷))

Proof of Theorem natffn

Dummy variables 𝑥 𝑓 𝑦 𝑎 𝑔 ℎ 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		natrcl.1	. . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷)
2		eqid 2765	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2765	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2765	. . 3 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
5		eqid 2765	. . 3 ⊢ (comp‘𝐷) = (comp‘𝐷)
6	1, 2, 3, 4, 5	natfval 18023	. 2 ⊢ 𝑁 = (𝑓 ∈ (𝐶 Func 𝐷), 𝑔 ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (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 7449	. . . . . . 7 ⊢ ((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∈ V
8	7	rgenw 3085	. . . . . 6 ⊢ ∀𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∈ V
9		ixpexg 8922	. . . . . 6 ⊢ (∀𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∈ V → X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∈ V)
10	8, 9	ax-mp 5	. . . . 5 ⊢ X𝑥 ∈ (Base‘𝐶)((𝑟‘𝑥)(Hom ‘𝐷)(𝑠‘𝑥)) ∈ V
11	10	rabex 5311	. . . 4 ⊢ {𝑎 ∈ X𝑥 ∈ (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 5276	. . 3 ⊢ ⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (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 5276	. 2 ⊢ ⦋(1st ‘𝑓) / 𝑟⦌⦋(1st ‘𝑔) / 𝑠⦌{𝑎 ∈ X𝑥 ∈ (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 8069	1 ⊢ 𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷))

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418  Vcvv 3457  ⦋csb 3854  ⟨cop 4597   × cxp 5661   Fn wfn 6535  ‘cfv 6540  (class class class)co 7416  1^st c1st 7986  2^nd c2nd 7987  Xcixp 8897  Basecbs 17286  Hom chom 17338  compcco 17339   Func cfunc 17928   Nat cnat 18018

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-ixp 8898  df-func 17932  df-nat 18020

This theorem is used by:  fuchom  18038