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Theorem catcfuccl 17363

Description: The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)

**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
catcfuccl	⊢ (𝜑 → 𝑄 ∈ 𝐵)

Proof of Theorem catcfuccl 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 2821	. . . . 5 ⊢ (𝑋 Func 𝑌) = (𝑋 Func 𝑌)
3		eqid 2821	. . . . 5 ⊢ (𝑋 Nat 𝑌) = (𝑋 Nat 𝑌)
4		eqid 2821	. . . . 5 ⊢ (Base‘𝑋) = (Base‘𝑋)
5		eqid 2821	. . . . 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 17351	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
11	6, 10	eleqtrd 2915	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
12	11	elin2d 4176	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Cat)
13		catcfuccl.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14	13, 10	eleqtrd 2915	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑈 ∩ Cat))
15	14	elin2d 4176	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ Cat)
16		eqidd 2822	. . . . 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 17222	. . . 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 16483	. . . . . . 7 ⊢ Base = Slot 1
19		catcfuccl.1	. . . . . . . 8 ⊢ (𝜑 → ω ∈ 𝑈)
20	9, 19	wunndx 16498	. . . . . . 7 ⊢ (𝜑 → ndx ∈ 𝑈)
21	18, 9, 20	wunstr 16501	. . . . . 6 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
22	11	elin1d 4175	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
23	14	elin1d 4175	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
24	9, 22, 23	wunfunc 17163	. . . . . 6 ⊢ (𝜑 → (𝑋 Func 𝑌) ∈ 𝑈)
25	9, 21, 24	wunop 10138	. . . . 5 ⊢ (𝜑 → ⟨(Base‘ndx), (𝑋 Func 𝑌)⟩ ∈ 𝑈)
26		df-hom 16583	. . . . . . 7 ⊢ Hom = Slot ;14
27	26, 9, 20	wunstr 16501	. . . . . 6 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
28	9, 22, 23	wunnat 17220	. . . . . 6 ⊢ (𝜑 → (𝑋 Nat 𝑌) ∈ 𝑈)
29	9, 27, 28	wunop 10138	. . . . 5 ⊢ (𝜑 → ⟨(Hom ‘ndx), (𝑋 Nat 𝑌)⟩ ∈ 𝑈)
30		df-cco 16584	. . . . . . 7 ⊢ comp = Slot ;15
31	30, 9, 20	wunstr 16501	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
32	9, 24, 24	wunxp 10140	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) ∈ 𝑈)
33	9, 32, 24	wunxp 10140	. . . . . . 7 ⊢ (𝜑 → (((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌)) ∈ 𝑈)
34	30, 9, 23	wunstr 16501	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑌) ∈ 𝑈)
35	9, 34	wunrn 10145	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
36	9, 35	wununi 10122	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑌) ∈ 𝑈)
37	9, 36	wunrn 10145	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑌) ∈ 𝑈)
38	9, 37	wununi 10122	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
39	9, 38	wunpw 10123	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
40	18, 9, 22	wunstr 16501	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
41	9, 39, 40	wunmap 10142	. . . . . . . 8 ⊢ (𝜑 → (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ∈ 𝑈)
42	9, 28	wunrn 10145	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑋 Nat 𝑌) ∈ 𝑈)
43	9, 42	wununi 10122	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran (𝑋 Nat 𝑌) ∈ 𝑈)
44	9, 43, 43	wunxp 10140	. . . . . . . 8 ⊢ (𝜑 → (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)) ∈ 𝑈)
45	9, 41, 44	wunpm 10141	. . . . . . 7 ⊢ (𝜑 → ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))) ∈ 𝑈)
46		fvex 6678	. . . . . . . . . . 11 ⊢ (1^st ‘𝑣) ∈ V
47		fvex 6678	. . . . . . . . . . . . . 14 ⊢ (2^nd ‘𝑣) ∈ V
48		ovex 7183	. . . . . . . . . . . . . . . . 17 ⊢ (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ∈ V
49		ovex 7183	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 Nat 𝑌) ∈ V
50	49	rnex 7611	. . . . . . . . . . . . . . . . . . 19 ⊢ ran (𝑋 Nat 𝑌) ∈ V
51	50	uniex 7461	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ ran (𝑋 Nat 𝑌) ∈ V
52	51, 51	xpex 7470	. . . . . . . . . . . . . . . . 17 ⊢ (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)) ∈ V
53		eqid 2821	. . . . . . . . . . . . . . . . . . . . 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 7186	. . . . . . . . . . . . . . . . . . . . . . . 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 7186	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥)) ⊆ ∪ ran (comp‘𝑌)
56		rnss 5804	. . . . . . . . . . . . . . . . . . . . . . . . 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 4853	. . . . . . . . . . . . . . . . . . . . . . . . 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 3976	. . . . . . . . . . . . . . . . . . . . . . 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 7183	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ∈ V
61	60	elpw 4546	. . . . . . . . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . . . . . . . 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 6871	. . . . . . . . . . . . . . . . . . . 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 6678	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (comp‘𝑌) ∈ V
66	65	rnex 7611	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ran (comp‘𝑌) ∈ V
67	66	uniex 7461	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ ran (comp‘𝑌) ∈ V
68	67	rnex 7611	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ran ∪ ran (comp‘𝑌) ∈ V
69	68	uniex 7461	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∪ ran ∪ ran (comp‘𝑌) ∈ V
70	69	pwex 5274	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ V
71		fvex 6678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝑋) ∈ V
72	70, 71	elmap 8429	. . . . . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . . . . 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 3151	. . . . . . . . . . . . . . . . . 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 2821	. . . . . . . . . . . . . . . . . . 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 7760	. . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . 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 7186	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔(𝑋 Nat 𝑌)ℎ) ⊆ ∪ ran (𝑋 Nat 𝑌)
79		ovssunirn 7186	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓(𝑋 Nat 𝑌)𝑔) ⊆ ∪ ran (𝑋 Nat 𝑌)
80		xpss12 5565	. . . . . . . . . . . . . . . . . 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 690	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))
82		elpm2r 8418	. . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . . 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 3787	. . . . . . . . . . . . . . 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 4365	. . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . 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 3787	. . . . . . . . . . . 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 4365	. . . . . . . . . . . 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 234	. . . . . . . . . . 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 3151	. . . . . . . . 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 2821	. . . . . . . . . 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 7760	. . . . . . . . 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 232	. . . . . . . 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 10143	. . . . . 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 10138	. . . . 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 10127	. . . 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 2913	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝑈)
101	1, 12, 15	fuccat 17234	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
102	100, 101	elind 4171	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑈 ∩ Cat))
103	102, 10	eleqtrrd 2916	1 ⊢ (𝜑 → 𝑄 ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3495 [wsbc 3772 ⦋csb 3883 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 {ctp 4565 ⟨cop 4567 ∪ cuni 4832 ↦ cmpt 5139 × cxp 5548 ran crn 5551 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ∈ cmpo 7152 ωcom 7574 1^st c1st 7681 2^nd c2nd 7682 ↑_m cmap 8400 ↑_pm cpm 8401 WUnicwun 10116 1c1 10532 4c4 11688 5c5 11689 cdc 12092 ndxcnx 16474 Basecbs 16477 Hom chom 16570 compcco 16571 Catccat 16929 Func cfunc 17118 Nat cnat 17205 FuncCat cfuc 17206 CatCatccatc 17348

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-wun 10118 df-ni 10288 df-pli 10289 df-mi 10290 df-lti 10291 df-plpq 10324 df-mpq 10325 df-ltpq 10326 df-enq 10327 df-nq 10328 df-erq 10329 df-plq 10330 df-mq 10331 df-1nq 10332 df-rq 10333 df-ltnq 10334 df-np 10397 df-plp 10399 df-ltp 10401 df-enr 10471 df-nr 10472 df-c 10537 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-hom 16583 df-cco 16584 df-cat 16933 df-cid 16934 df-func 17122 df-nat 17207 df-fuc 17208 df-catc 17349

This theorem is referenced by: (None)

W3C validator