fucco - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > fucco		Structured version Visualization version GIF version

Theorem fucco 17911

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)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘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 2732	. . . 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 17897	. . . . . . . 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 495	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
10		funcrcl 17809	. . . . . 6 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
12	11	simpld 495	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
13	11	simprd 496	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
14		fucco.x	. . . 4 ⊢ ∙ = (comp‘𝑄)
15	1, 2, 3, 4, 5, 12, 13, 14	fuccofval 17907	. . 3 ⊢ (𝜑 → ∙ = (𝑣 ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)), ℎ ∈ (𝐶 Func 𝐷) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))
16		fvexd 6903	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1^st ‘𝑣) ∈ V)
17		simprl 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → 𝑣 = ⟨𝐹, 𝐺⟩)
18	17	fveq2d 6892	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1^st ‘𝑣) = (1^st ‘⟨𝐹, 𝐺⟩))
19		op1stg 7983	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘⟨𝐹, 𝐺⟩) = 𝐹)
20	8, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (1^st ‘⟨𝐹, 𝐺⟩) = 𝐹)
21	20	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1^st ‘⟨𝐹, 𝐺⟩) = 𝐹)
22	18, 21	eqtrd 2772	. . . 4 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → (1^st ‘𝑣) = 𝐹)
23		fvexd 6903	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2^nd ‘𝑣) ∈ V)
24	17	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → 𝑣 = ⟨𝐹, 𝐺⟩)
25	24	fveq2d 6892	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2^nd ‘𝑣) = (2^nd ‘⟨𝐹, 𝐺⟩))
26		op2ndg 7984	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (2^nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
27	8, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → (2^nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
28	27	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2^nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
29	25, 28	eqtrd 2772	. . . . 5 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → (2^nd ‘𝑣) = 𝐺)
30		simpr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
31		simprr 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → ℎ = 𝐻)
32	31	ad2antrr 724	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ℎ = 𝐻)
33	30, 32	oveq12d 7423	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑔𝑁ℎ) = (𝐺𝑁𝐻))
34		simplr 767	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
35	34, 30	oveq12d 7423	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑓𝑁𝑔) = (𝐹𝑁𝐺))
36	34	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1^st ‘𝑓) = (1^st ‘𝐹))
37	36	fveq1d 6890	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1^st ‘𝑓)‘𝑥) = ((1^st ‘𝐹)‘𝑥))
38	30	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1^st ‘𝑔) = (1^st ‘𝐺))
39	38	fveq1d 6890	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1^st ‘𝑔)‘𝑥) = ((1^st ‘𝐺)‘𝑥))
40	37, 39	opeq12d 4880	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ = ⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩)
41	32	fveq2d 6892	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (1^st ‘ℎ) = (1^st ‘𝐻))
42	41	fveq1d 6890	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((1^st ‘ℎ)‘𝑥) = ((1^st ‘𝐻)‘𝑥))
43	40, 42	oveq12d 7423	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥)) = (⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥)))
44	43	oveqd 7422	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)) = ((𝑏‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑎‘𝑥)))
45	44	mpteq2dv 5249	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑎‘𝑥))))
46	33, 35, 45	mpoeq123dv 7480	. . . . 5 ⊢ ((((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) ∧ 𝑔 = 𝐺) → (𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑎‘𝑥)))))
47	23, 29, 46	csbied2 3932	. . . 4 ⊢ (((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) ∧ 𝑓 = 𝐹) → ⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑎‘𝑥)))))
48	16, 22, 47	csbied2 3932	. . 3 ⊢ ((𝜑 ∧ (𝑣 = ⟨𝐹, 𝐺⟩ ∧ ℎ = 𝐻)) → ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩ · ((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑎‘𝑥)))))
49		opelxpi 5712	. . . 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 17897	. . . . 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 496	. . 3 ⊢ (𝜑 → 𝐻 ∈ (𝐶 Func 𝐷))
55		ovex 7438	. . . . 5 ⊢ (𝐺𝑁𝐻) ∈ V
56		ovex 7438	. . . . 5 ⊢ (𝐹𝑁𝐺) ∈ V
57	55, 56	mpoex 8062	. . . 4 ⊢ (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑎‘𝑥)))) ∈ V
58	57	a1i 11	. . 3 ⊢ (𝜑 → (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑎‘𝑥)))) ∈ V)
59	15, 48, 50, 54, 58	ovmpod 7556	. 2 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ∙ 𝐻) = (𝑏 ∈ (𝐺𝑁𝐻), 𝑎 ∈ (𝐹𝑁𝐺) ↦ (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑎‘𝑥)))))
60		simprl 769	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → 𝑏 = 𝑆)
61	60	fveq1d 6890	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → (𝑏‘𝑥) = (𝑆‘𝑥))
62		simprr 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → 𝑎 = 𝑅)
63	62	fveq1d 6890	. . . 4 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → (𝑎‘𝑥) = (𝑅‘𝑥))
64	61, 63	oveq12d 7423	. . 3 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → ((𝑏‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑎‘𝑥)) = ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥)))
65	64	mpteq2dv 5249	. 2 ⊢ ((𝜑 ∧ (𝑏 = 𝑆 ∧ 𝑎 = 𝑅)) → (𝑥 ∈ 𝐴 ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑎‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥))))
66	4	fvexi 6902	. . . 4 ⊢ 𝐴 ∈ V
67	66	mptex 7221	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥))) ∈ V
68	67	a1i 11	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥))) ∈ V)
69	59, 65, 51, 6, 68	ovmpod 7556	1 ⊢ (𝜑 → (𝑆(⟨𝐹, 𝐺⟩ ∙ 𝐻)𝑅) = (𝑥 ∈ 𝐴 ↦ ((𝑆‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐻)‘𝑥))(𝑅‘𝑥))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⦋csb 3892 ⟨cop 4633 ↦ cmpt 5230 × cxp 5673 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 1^st c1st 7969 2^nd c2nd 7970 Basecbs 17140 compcco 17205 Catccat 17604 Func cfunc 17800 Nat cnat 17888 FuncCat cfuc 17889

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-hom 17217 df-cco 17218 df-func 17804 df-nat 17890 df-fuc 17891

This theorem is referenced by: fuccoval 17912 fuccocl 17913 fuclid 17915 fucrid 17916 fucass 17917 fucsect 17921 curfcl 18181

W3C validator