Theorem fucco 16622

Description: Value of the composition of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
fucco.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fucco.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
fucco.a	⊢ 𝐴 = (Base‘𝐶)
fucco.o	⊢ · = (comp‘𝐷)
fucco.x	⊢ ∙ = (comp‘𝑄)
fucco.f	⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
fucco.g	⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))

Assertion

Ref	Expression
fucco	⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))))

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝑅   𝑥,𝑆   𝑥,𝐶   𝑥,𝐷   𝑥, ·   𝑥,𝐹   𝑥,𝐺   𝑥,𝐻

Allowed substitution hints:   𝑄(𝑥)   ∙ (𝑥)   𝑁(𝑥)

Proof of Theorem fucco

Dummy variables 𝑎 𝑏 𝑓 𝑔 ℎ 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucco.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2		eqid 2622	. . . 4 ⊢ (𝐶 Func 𝐷) = (𝐶 Func 𝐷)
3		fucco.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4		fucco.a	. . . 4 ⊢ 𝐴 = (Base‘𝐶)
5		fucco.o	. . . 4 ⊢ · = (comp‘𝐷)
6		fucco.f	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
7	3	natrcl 16610	. . . . . . . 8 ⊢ (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
8	6, 7	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
9	8	simpld 475	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
10		funcrcl 16523	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
12	11	simpld 475	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
13	11	simprd 479	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
14		fucco.x	. . . 4 ⊢ ∙ = (comp‘𝑄)
15	1, 2, 3, 4, 5, 12, 13, 14	fuccofval 16619	. . 3 ⊢ (𝜑 → ∙ = (𝑣 ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)), ℎ ∈ (𝐶 Func 𝐷) ↦ ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))
16		fvexd 6203	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘𝑣) ∈ V)
17		simprl 794	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → 𝑣 = ⟨𝐹, 𝐺⟩)
18	17	fveq2d 6195	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘𝑣) = (1st ‘⟨𝐹, 𝐺⟩))
19		op1stg 7180	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
20	8, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
21	20	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
22	18, 21	eqtrd 2656	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1st ‘𝑣) = 𝐹)
23		fvexd 6203	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘𝑣) ∈ V)
24	17	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → 𝑣 = ⟨𝐹, 𝐺⟩)
25	24	fveq2d 6195	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘𝑣) = (2nd ‘⟨𝐹, 𝐺⟩))
26		op2ndg 7181	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
27	8, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
28	27	ad2antrr 762	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29	25, 28	eqtrd 2656	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2nd ‘𝑣) = 𝐺)
30		simpr 477	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
31		simprr 796	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → ℎ = 𝐻)
32	31	ad2antrr 762	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ℎ = 𝐻)
33	30, 32	oveq12d 6668	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔𝑁ℎ) = (𝐺𝑁𝐻))
34		simplr 792	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
35	34, 30	oveq12d 6668	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓𝑁𝑔) = (𝐹𝑁𝐺))
36	34	fveq2d 6195	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st ‘𝑓) = (1st ‘𝐹))
37	36	fveq1d 6193	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st ‘𝑓)‘𝑥) = ((1st ‘𝐹)‘𝑥))
38	30	fveq2d 6195	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st ‘𝑔) = (1st ‘𝐺))
39	38	fveq1d 6193	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st ‘𝑔)‘𝑥) = ((1st ‘𝐺)‘𝑥))
40	37, 39	opeq12d 4410	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ = ⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩)
41	32	fveq2d 6195	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1st ‘ℎ) = (1st ‘𝐻))
42	41	fveq1d 6193	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1st ‘ℎ)‘𝑥) = ((1st ‘𝐻)‘𝑥))
43	40, 42	oveq12d 6668	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥)) = (⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥)))
44	43	oveqd 6667	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)) = ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))
45	44	mpteq2dv 4745	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥))))
46	33, 35, 45	mpt2eq123dv 6717	. . . . 5 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))))
47	23, 29, 46	csbied2 3561	. . . 4 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → ⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))))
48	16, 22, 47	csbied2 3561	. . 3 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → ⦋(1st ‘𝑣) / 𝑓⦌⦋(2nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝑓)‘𝑥), ((1st ‘𝑔)‘𝑥)⟩ · ((1st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))))
49		opelxpi 5148	. . . 4 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → ⟨𝐹, 𝐺⟩ ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)))
50	8, 49	syl 17	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)))
51		fucco.g	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ (𝐺𝑁𝐻))
52	3	natrcl 16610	. . . . 5 ⊢ (𝑆 ∈ (𝐺𝑁𝐻) → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
53	51, 52	syl 17	. . . 4 ⊢ (𝜑 → (𝐺 ∈ (𝐶 Func 𝐷) ∧ 𝐻 ∈ (𝐶 Func 𝐷)))
54	53	simprd 479	. . 3 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
55		ovex 6678	. . . . 5 ⊢ (𝐺𝑁𝐻) ∈ V
56		ovex 6678	. . . . 5 ⊢ (𝐹𝑁𝐺) ∈ V
57	55, 56	mpt2ex 7247	. . . 4 ⊢ (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))) ∈ V
58	57	a1i 11	. . 3 ⊢ (𝜑 → (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))) ∈ V)
59	15, 48, 50, 54, 58	ovmpt2d 6788	. 2 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∙ 𝐻) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)))))
60		simprl 794	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → 𝑏 = 𝑆)
61	60	fveq1d 6193	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → (𝑏‘𝑥) = (𝑆‘𝑥))
62		simprr 796	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → 𝑎 = 𝑅)
63	62	fveq1d 6193	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → (𝑎‘𝑥) = (𝑅‘𝑥))
64	61, 63	oveq12d 6668	. . 3 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥)) = ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥)))
65	64	mpteq2dv 4745	. 2 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑎‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
66		fvex 6201	. . . . 5 ⊢ (Base‘𝐶) ∈ V
67	4, 66	eqeltri 2697	. . . 4 ⊢ 𝐴 ∈ V
68	67	mptex 6486	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))) ∈ V
69	68	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))) ∈ V)
70	59, 65, 51, 6, 69	ovmpt2d 6788	1 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)⟩ · ((1st ‘𝐻)‘𝑥))(𝑅‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⦋csb 3533  ⟨cop 4183   ↦ cmpt 4729   × cxp 5112  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167  Basecbs 15857  compcco 15953  Catccat 16325   Func cfunc 16514   Nat cnat 16601   FuncCat cfuc 16602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-func 16518  df-nat 16603  df-fuc 16604

This theorem is referenced by:  fuccoval  16623  fuccocl  16624  fuclid  16626  fucrid  16627  fucass  16628  fucsect  16632  curfcl  16872