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Theorem catcfucclOLD 18102

Description: Obsolete proof of catcfuccl 18101 as of 14-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
catcfuccl.c	⊢ 𝐶 = (CatCat‘𝑈)
catcfuccl.b	⊢ 𝐵 = (Base‘𝐶)
catcfuccl.o	⊢ 𝑄 = (𝑋 FuncCat 𝑌)
catcfuccl.u	⊢ (𝜑 → 𝑈 ∈ WUni)
catcfuccl.1	⊢ (𝜑 → ω ∈ 𝑈)
catcfuccl.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
catcfuccl.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
catcfucclOLD	⊢ (𝜑 → 𝑄 ∈ 𝐵)

Proof of Theorem catcfucclOLD Dummy variables 𝑎 𝑏 𝑓 𝑔 ℎ 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		catcfuccl.o	. . . . 5 ⊢ 𝑄 = (𝑋 FuncCat 𝑌)
2		eqid 2727	. . . . 5 ⊢ (𝑋 Func 𝑌) = (𝑋 Func 𝑌)
3		eqid 2727	. . . . 5 ⊢ (𝑋 Nat 𝑌) = (𝑋 Nat 𝑌)
4		eqid 2727	. . . . 5 ⊢ (Base‘𝑋) = (Base‘𝑋)
5		eqid 2727	. . . . 5 ⊢ (comp‘𝑌) = (comp‘𝑌)
6		catcfuccl.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		catcfuccl.c	. . . . . . . 8 ⊢ 𝐶 = (CatCat‘𝑈)
8		catcfuccl.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
9		catcfuccl.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ WUni)
10	7, 8, 9	catcbas 18083	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
11	6, 10	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
12	11	elin2d 4195	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Cat)
13		catcfuccl.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14	13, 10	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑈 ∩ Cat))
15	14	elin2d 4195	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ Cat)
16		eqidd 2728	. . . . 5 ⊢ (𝜑 → (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) = (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))
17	1, 2, 3, 4, 5, 12, 15, 16	fucval 17942	. . . 4 ⊢ (𝜑 → 𝑄 = {⟨(Base‘ndx), (𝑋 Func 𝑌)⟩, ⟨(Hom ‘ndx), (𝑋 Nat 𝑌)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩})
18		df-base 17174	. . . . . . 7 ⊢ Base = Slot 1
19		catcfuccl.1	. . . . . . . 8 ⊢ (𝜑 → ω ∈ 𝑈)
20	9, 19	wunndx 17157	. . . . . . 7 ⊢ (𝜑 → ndx ∈ 𝑈)
21	18, 9, 20	wunstr 17150	. . . . . 6 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
22	11	elin1d 4194	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
23	14	elin1d 4194	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
24	9, 22, 23	wunfunc 17880	. . . . . 6 ⊢ (𝜑 → (𝑋 Func 𝑌) ∈ 𝑈)
25	9, 21, 24	wunop 10739	. . . . 5 ⊢ (𝜑 → ⟨(Base‘ndx), (𝑋 Func 𝑌)⟩ ∈ 𝑈)
26		df-hom 17250	. . . . . . 7 ⊢ Hom = Slot ;14
27	26, 9, 20	wunstr 17150	. . . . . 6 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
28	9, 22, 23	wunnat 17939	. . . . . 6 ⊢ (𝜑 → (𝑋 Nat 𝑌) ∈ 𝑈)
29	9, 27, 28	wunop 10739	. . . . 5 ⊢ (𝜑 → ⟨(Hom ‘ndx), (𝑋 Nat 𝑌)⟩ ∈ 𝑈)
30		df-cco 17251	. . . . . . 7 ⊢ comp = Slot ;15
31	30, 9, 20	wunstr 17150	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
32	9, 24, 24	wunxp 10741	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) ∈ 𝑈)
33	9, 32, 24	wunxp 10741	. . . . . . 7 ⊢ (𝜑 → (((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌)) ∈ 𝑈)
34	30, 9, 23	wunstr 17150	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑌) ∈ 𝑈)
35	9, 34	wunrn 10746	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
36	9, 35	wununi 10723	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑌) ∈ 𝑈)
37	9, 36	wunrn 10746	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑌) ∈ 𝑈)
38	9, 37	wununi 10723	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
39	9, 38	wunpw 10724	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
40	18, 9, 22	wunstr 17150	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
41	9, 39, 40	wunmap 10743	. . . . . . . 8 ⊢ (𝜑 → (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ∈ 𝑈)
42	9, 28	wunrn 10746	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑋 Nat 𝑌) ∈ 𝑈)
43	9, 42	wununi 10723	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran (𝑋 Nat 𝑌) ∈ 𝑈)
44	9, 43, 43	wunxp 10741	. . . . . . . 8 ⊢ (𝜑 → (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)) ∈ 𝑈)
45	9, 41, 44	wunpm 10742	. . . . . . 7 ⊢ (𝜑 → ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))) ∈ 𝑈)
46		fvex 6904	. . . . . . . . . . 11 ⊢ (1^st ‘𝑣) ∈ V
47		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (2^nd ‘𝑣) ∈ V
48		ovex 7447	. . . . . . . . . . . . . . . . 17 ⊢ (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ∈ V
49		ovex 7447	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 Nat 𝑌) ∈ V
50	49	rnex 7912	. . . . . . . . . . . . . . . . . . 19 ⊢ ran (𝑋 Nat 𝑌) ∈ V
51	50	uniex 7740	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ ran (𝑋 Nat 𝑌) ∈ V
52	51, 51	xpex 7749	. . . . . . . . . . . . . . . . 17 ⊢ (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)) ∈ V
53		eqid 2727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))) = (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))
54		ovssunirn 7450	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ⊆ ∪ ran (⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))
55		ovssunirn 7450	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥)) ⊆ ∪ ran (comp‘𝑌)
56		rnss 5935	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥)) ⊆ ∪ ran (comp‘𝑌) → ran (⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥)) ⊆ ran ∪ ran (comp‘𝑌))
57		uniss 4911	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ran (⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥)) ⊆ ran ∪ ran (comp‘𝑌) → ∪ ran (⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥)) ⊆ ∪ ran ∪ ran (comp‘𝑌))
58	55, 56, 57	mp2b 10	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ ran (⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥)) ⊆ ∪ ran ∪ ran (comp‘𝑌)
59	54, 58	sstri 3987	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ⊆ ∪ ran ∪ ran (comp‘𝑌)
60		ovex 7447	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ∈ V
61	60	elpw 4602	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌) ↔ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ⊆ ∪ ran ∪ ran (comp‘𝑌))
62	59, 61	mpbir 230	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌)
63	62	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ (Base‘𝑋) → ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌))
64	53, 63	fmpti 7116	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))):(Base‘𝑋)⟶𝒫 ∪ ran ∪ ran (comp‘𝑌)
65		fvex 6904	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (comp‘𝑌) ∈ V
66	65	rnex 7912	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ran (comp‘𝑌) ∈ V
67	66	uniex 7740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ ran (comp‘𝑌) ∈ V
68	67	rnex 7912	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ran ∪ ran (comp‘𝑌) ∈ V
69	68	uniex 7740	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∪ ran ∪ ran (comp‘𝑌) ∈ V
70	69	pwex 5374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ V
71		fvex 6904	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝑋) ∈ V
72	70, 71	elmap 8883	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↔ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))):(Base‘𝑋)⟶𝒫 ∪ ran ∪ ran (comp‘𝑌))
73	64, 72	mpbir 230	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋))
74	73	rgen2w 3061	. . . . . . . . . . . . . . . . . 18 ⊢ ∀𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ)∀𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔)(𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋))
75		eqid 2727	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) = (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))
76	75	fmpo 8066	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ)∀𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔)(𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))) ∈ (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↔ (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))):((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔))⟶(𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)))
77	74, 76	mpbi 229	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))):((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔))⟶(𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋))
78		ovssunirn 7450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔(𝑋 Nat 𝑌)ℎ) ⊆ ∪ ran (𝑋 Nat 𝑌)
79		ovssunirn 7450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓(𝑋 Nat 𝑌)𝑔) ⊆ ∪ ran (𝑋 Nat 𝑌)
80		xpss12 5687	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑔(𝑋 Nat 𝑌)ℎ) ⊆ ∪ ran (𝑋 Nat 𝑌) ∧ (𝑓(𝑋 Nat 𝑌)𝑔) ⊆ ∪ ran (𝑋 Nat 𝑌)) → ((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)))
81	78, 79, 80	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))
82		elpm2r 8857	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ∈ V ∧ (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)) ∈ V) ∧ ((𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))):((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔))⟶(𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ∧ ((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)))) → (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))))
83	48, 52, 77, 81, 82	mp4an 692	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)))
84	83	sbcth 3789	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘𝑣) ∈ V → [(2^nd ‘𝑣) / 𝑔](𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))))
85		sbcel1g 4409	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘𝑣) ∈ V → ([(2^nd ‘𝑣) / 𝑔](𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))) ↔ ⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)))))
86	84, 85	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘𝑣) ∈ V → ⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))))
87	47, 86	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)))
88	87	sbcth 3789	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑣) ∈ V → [(1^st ‘𝑣) / 𝑓]⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))))
89		sbcel1g 4409	. . . . . . . . . . . 12 ⊢ ((1^st ‘𝑣) ∈ V → ([(1^st ‘𝑣) / 𝑓]⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))) ↔ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)))))
90	88, 89	mpbid 231	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑣) ∈ V → ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))))
91	46, 90	ax-mp 5	. . . . . . . . . 10 ⊢ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)))
92	91	rgen2w 3061	. . . . . . . . 9 ⊢ ∀𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌))∀ℎ ∈ (𝑋 Func 𝑌)⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)))
93		eqid 2727	. . . . . . . . . 10 ⊢ (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) = (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))
94	93	fmpo 8066	. . . . . . . . 9 ⊢ (∀𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌))∀ℎ ∈ (𝑋 Func 𝑌)⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))) ↔ (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))):(((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌))⟶((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))))
95	92, 94	mpbi 229	. . . . . . . 8 ⊢ (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))):(((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌))⟶((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)))
96	95	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))):(((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌))⟶((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))))
97	9, 33, 45, 96	wunf 10744	. . . . . 6 ⊢ (𝜑 → (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))) ∈ 𝑈)
98	9, 31, 97	wunop 10739	. . . . 5 ⊢ (𝜑 → ⟨(comp‘ndx), (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩ ∈ 𝑈)
99	9, 25, 29, 98	wuntp 10728	. . . 4 ⊢ (𝜑 → {⟨(Base‘ndx), (𝑋 Func 𝑌)⟩, ⟨(Hom ‘ndx), (𝑋 Nat 𝑌)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)), ℎ ∈ (𝑋 Func 𝑌) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔(𝑋 Nat 𝑌)ℎ), 𝑎 ∈ (𝑓(𝑋 Nat 𝑌)𝑔) ↦ (𝑥 ∈ (Base‘𝑋) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)))))⟩} ∈ 𝑈)
100	17, 99	eqeltrd 2828	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝑈)
101	1, 12, 15	fuccat 17955	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
102	100, 101	elind 4190	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑈 ∩ Cat))
103	102, 10	eleqtrrd 2831	1 ⊢ (𝜑 → 𝑄 ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∀wral 3056 Vcvv 3469 [wsbc 3774 ⦋csb 3889 ∩ cin 3943 ⊆ wss 3944 𝒫 cpw 4598 {ctp 4628 ⟨cop 4630 ∪ cuni 4903 ↦ cmpt 5225 × cxp 5670 ran crn 5673 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 ωcom 7864 1^st c1st 7985 2^nd c2nd 7986 ↑_m cmap 8838 ↑_pm cpm 8839 WUnicwun 10717 1c1 11133 4c4 12293 5c5 12294 cdc 12701 ndxcnx 17155 Basecbs 17173 Hom chom 17237 compcco 17238 Catccat 17637 Func cfunc 17833 Nat cnat 17924 FuncCat cfuc 17925 CatCatccatc 18080

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-omul 8485 df-er 8718 df-ec 8720 df-qs 8724 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-wun 10719 df-ni 10889 df-pli 10890 df-mi 10891 df-lti 10892 df-plpq 10925 df-mpq 10926 df-ltpq 10927 df-enq 10928 df-nq 10929 df-erq 10930 df-plq 10931 df-mq 10932 df-1nq 10933 df-rq 10934 df-ltnq 10935 df-np 10998 df-plp 11000 df-ltp 11002 df-enr 11072 df-nr 11073 df-c 11138 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-struct 17109 df-slot 17144 df-ndx 17156 df-base 17174 df-hom 17250 df-cco 17251 df-cat 17641 df-cid 17642 df-func 17837 df-nat 17926 df-fuc 17927 df-catc 18081

This theorem is referenced by: (None)

W3C validator