catcfuccl - Metamath Proof Explorer

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

Theorem catcfuccl 18043

Description: The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)

**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 2737	. . . . 5 ⊢ (𝑋 Func 𝑌) = (𝑋 Func 𝑌)
3		eqid 2737	. . . . 5 ⊢ (𝑋 Nat 𝑌) = (𝑋 Nat 𝑌)
4		eqid 2737	. . . . 5 ⊢ (Base‘𝑋) = (Base‘𝑋)
5		eqid 2737	. . . . 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 18026	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
11	6, 10	eleqtrd 2839	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
12	11	elin2d 4146	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Cat)
13		catcfuccl.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14	13, 10	eleqtrd 2839	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑈 ∩ Cat))
15	14	elin2d 4146	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ Cat)
16		eqidd 2738	. . . . 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 17886	. . . 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		baseid 17140	. . . . . . 7 ⊢ Base = Slot (Base‘ndx)
19		catcfuccl.1	. . . . . . . 8 ⊢ (𝜑 → ω ∈ 𝑈)
20	9, 19	wunndx 17123	. . . . . . 7 ⊢ (𝜑 → ndx ∈ 𝑈)
21	18, 9, 20	wunstr 17116	. . . . . 6 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
22	7, 8, 9, 6	catcbascl 18037	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
23	7, 8, 9, 13	catcbascl 18037	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
24	9, 22, 23	wunfunc 17826	. . . . . 6 ⊢ (𝜑 → (𝑋 Func 𝑌) ∈ 𝑈)
25	9, 21, 24	wunop 10634	. . . . 5 ⊢ (𝜑 → ⟨(Base‘ndx), (𝑋 Func 𝑌)⟩ ∈ 𝑈)
26		homid 17333	. . . . . . 7 ⊢ Hom = Slot (Hom ‘ndx)
27	26, 9, 20	wunstr 17116	. . . . . 6 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
28	9, 22, 23	wunnat 17884	. . . . . 6 ⊢ (𝜑 → (𝑋 Nat 𝑌) ∈ 𝑈)
29	9, 27, 28	wunop 10634	. . . . 5 ⊢ (𝜑 → ⟨(Hom ‘ndx), (𝑋 Nat 𝑌)⟩ ∈ 𝑈)
30		ccoid 17335	. . . . . . 7 ⊢ comp = Slot (comp‘ndx)
31	30, 9, 20	wunstr 17116	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
32	9, 24, 24	wunxp 10636	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) ∈ 𝑈)
33	9, 32, 24	wunxp 10636	. . . . . . 7 ⊢ (𝜑 → (((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌)) ∈ 𝑈)
34	7, 8, 9, 13	catcccocl 18041	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑌) ∈ 𝑈)
35	9, 34	wunrn 10641	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
36	9, 35	wununi 10618	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑌) ∈ 𝑈)
37	9, 36	wunrn 10641	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑌) ∈ 𝑈)
38	9, 37	wununi 10618	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
39	9, 38	wunpw 10619	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
40	7, 8, 9, 6	catcbaselcl 18039	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
41	9, 39, 40	wunmap 10638	. . . . . . . 8 ⊢ (𝜑 → (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ∈ 𝑈)
42	9, 28	wunrn 10641	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑋 Nat 𝑌) ∈ 𝑈)
43	9, 42	wununi 10618	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran (𝑋 Nat 𝑌) ∈ 𝑈)
44	9, 43, 43	wunxp 10636	. . . . . . . 8 ⊢ (𝜑 → (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)) ∈ 𝑈)
45	9, 41, 44	wunpm 10637	. . . . . . 7 ⊢ (𝜑 → ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))) ∈ 𝑈)
46		fvex 6845	. . . . . . . . . . 11 ⊢ (1^st ‘𝑣) ∈ V
47		fvex 6845	. . . . . . . . . . . . . 14 ⊢ (2^nd ‘𝑣) ∈ V
48		ovex 7391	. . . . . . . . . . . . . . . . 17 ⊢ (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ∈ V
49		ovex 7391	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 Nat 𝑌) ∈ V
50	49	rnex 7852	. . . . . . . . . . . . . . . . . . 19 ⊢ ran (𝑋 Nat 𝑌) ∈ V
51	50	uniex 7686	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ ran (𝑋 Nat 𝑌) ∈ V
52	51, 51	xpex 7698	. . . . . . . . . . . . . . . . 17 ⊢ (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)) ∈ V
53		eqid 2737	. . . . . . . . . . . . . . . . . . . . 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 7394	. . . . . . . . . . . . . . . . . . . . . . . 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 7394	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥)) ⊆ ∪ ran (comp‘𝑌)
56		rnss 5886	. . . . . . . . . . . . . . . . . . . . . . . . 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 4859	. . . . . . . . . . . . . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . . . . . . . . . . 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 7391	. . . . . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . . . . 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 7056	. . . . . . . . . . . . . . . . . . . 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 6845	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (comp‘𝑌) ∈ V
66	65	rnex 7852	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ran (comp‘𝑌) ∈ V
67	66	uniex 7686	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ ran (comp‘𝑌) ∈ V
68	67	rnex 7852	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ran ∪ ran (comp‘𝑌) ∈ V
69	68	uniex 7686	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∪ ran ∪ ran (comp‘𝑌) ∈ V
70	69	pwex 5315	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ V
71		fvex 6845	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝑋) ∈ V
72	70, 71	elmap 8810	. . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . 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 3057	. . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . 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 8012	. . . . . . . . . . . . . . . . . 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 230	. . . . . . . . . . . . . . . . 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 7394	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔(𝑋 Nat 𝑌)ℎ) ⊆ ∪ ran (𝑋 Nat 𝑌)
79		ovssunirn 7394	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓(𝑋 Nat 𝑌)𝑔) ⊆ ∪ ran (𝑋 Nat 𝑌)
80		xpss12 5637	. . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))
82		elpm2r 8783	. . . . . . . . . . . . . . . . 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 694	. . . . . . . . . . . . . . . 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 3744	. . . . . . . . . . . . . . 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 4357	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 3744	. . . . . . . . . . . 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 4357	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 3057	. . . . . . . . 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 2737	. . . . . . . . . 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 8012	. . . . . . . . 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 230	. . . . . . . 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 10639	. . . . . 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 10634	. . . . 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 10623	. . . 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 2837	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝑈)
101	1, 12, 15	fuccat 17898	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
102	100, 101	elind 4141	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑈 ∩ Cat))
103	102, 10	eleqtrrd 2840	1 ⊢ (𝜑 → 𝑄 ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 [wsbc 3729 ⦋csb 3838 ∩ cin 3889 ⊆ wss 3890 𝒫 cpw 4542 {ctp 4572 ⟨cop 4574 ∪ cuni 4851 ↦ cmpt 5167 × cxp 5620 ran crn 5623 ⟶wf 6486 ‘cfv 6490 (class class class)co 7358 ∈ cmpo 7360 ωcom 7808 1^st c1st 7931 2^nd c2nd 7932 ↑_m cmap 8764 ↑_pm cpm 8765 WUnicwun 10612 ndxcnx 17121 Basecbs 17137 Hom chom 17189 compcco 17190 Catccat 17588 Func cfunc 17779 Nat cnat 17869 FuncCat cfuc 17870 CatCatccatc 18023

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-oadd 8400 df-omul 8401 df-er 8634 df-ec 8636 df-qs 8640 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-wun 10614 df-ni 10784 df-pli 10785 df-mi 10786 df-lti 10787 df-plpq 10820 df-mpq 10821 df-ltpq 10822 df-enq 10823 df-nq 10824 df-erq 10825 df-plq 10826 df-mq 10827 df-1nq 10828 df-rq 10829 df-ltnq 10830 df-np 10893 df-plp 10895 df-ltp 10897 df-enr 10967 df-nr 10968 df-c 11033 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12609 df-uz 12753 df-fz 13425 df-struct 17075 df-slot 17110 df-ndx 17122 df-base 17138 df-hom 17202 df-cco 17203 df-cat 17592 df-cid 17593 df-func 17783 df-nat 17871 df-fuc 17872 df-catc 18024

This theorem is referenced by: (None)

W3C validator