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Theorem catcfuccl 18322

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 2763	. . . . 5 ⊢ (𝑋 Func 𝑌) = (𝑋 Func 𝑌)
3		eqid 2763	. . . . 5 ⊢ (𝑋 Nat 𝑌) = (𝑋 Nat 𝑌)
4		eqid 2763	. . . . 5 ⊢ (Base‘𝑋) = (Base‘𝑋)
5		eqid 2763	. . . . 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 18305	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
11	6, 10	eleqtrd 2866	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
12	11	elin2d 4105	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ Cat)
13		catcfuccl.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
14	13, 10	eleqtrd 2866	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (𝑈 ∩ Cat))
15	14	elin2d 4105	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ Cat)
16		eqidd 2764	. . . . 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 18165	. . . 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 17419	. . . . . . 7 ⊢ Base = Slot (Base‘ndx)
19		catcfuccl.1	. . . . . . . 8 ⊢ (𝜑 → ω ∈ 𝑈)
20	9, 19	wunndx 17402	. . . . . . 7 ⊢ (𝜑 → ndx ∈ 𝑈)
21	18, 9, 20	wunstr 17395	. . . . . 6 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
22	7, 8, 9, 6	catcbascl 18316	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
23	7, 8, 9, 13	catcbascl 18316	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
24	9, 22, 23	wunfunc 18105	. . . . . 6 ⊢ (𝜑 → (𝑋 Func 𝑌) ∈ 𝑈)
25	9, 21, 24	wunop 10834	. . . . 5 ⊢ (𝜑 → ⟨(Base‘ndx), (𝑋 Func 𝑌)⟩ ∈ 𝑈)
26		homid 17612	. . . . . . 7 ⊢ Hom = Slot (Hom ‘ndx)
27	26, 9, 20	wunstr 17395	. . . . . 6 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
28	9, 22, 23	wunnat 18163	. . . . . 6 ⊢ (𝜑 → (𝑋 Nat 𝑌) ∈ 𝑈)
29	9, 27, 28	wunop 10834	. . . . 5 ⊢ (𝜑 → ⟨(Hom ‘ndx), (𝑋 Nat 𝑌)⟩ ∈ 𝑈)
30		ccoid 17614	. . . . . . 7 ⊢ comp = Slot (comp‘ndx)
31	30, 9, 20	wunstr 17395	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
32	9, 24, 24	wunxp 10836	. . . . . . . 8 ⊢ (𝜑 → ((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) ∈ 𝑈)
33	9, 32, 24	wunxp 10836	. . . . . . 7 ⊢ (𝜑 → (((𝑋 Func 𝑌) × (𝑋 Func 𝑌)) × (𝑋 Func 𝑌)) ∈ 𝑈)
34	7, 8, 9, 13	catcccocl 18320	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑌) ∈ 𝑈)
35	9, 34	wunrn 10841	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
36	9, 35	wununi 10818	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑌) ∈ 𝑈)
37	9, 36	wunrn 10841	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑌) ∈ 𝑈)
38	9, 37	wununi 10818	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
39	9, 38	wunpw 10819	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
40	7, 8, 9, 6	catcbaselcl 18318	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
41	9, 39, 40	wunmap 10838	. . . . . . . 8 ⊢ (𝜑 → (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ∈ 𝑈)
42	9, 28	wunrn 10841	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑋 Nat 𝑌) ∈ 𝑈)
43	9, 42	wununi 10818	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran (𝑋 Nat 𝑌) ∈ 𝑈)
44	9, 43, 43	wunxp 10836	. . . . . . . 8 ⊢ (𝜑 → (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)) ∈ 𝑈)
45	9, 41, 44	wunpm 10837	. . . . . . 7 ⊢ (𝜑 → ((𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ↑_pm (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))) ∈ 𝑈)
46		fvex 6921	. . . . . . . . . . 11 ⊢ (1^st ‘𝑣) ∈ V
47		fvex 6921	. . . . . . . . . . . . . 14 ⊢ (2^nd ‘𝑣) ∈ V
48		ovex 7476	. . . . . . . . . . . . . . . . 17 ⊢ (𝒫 ∪ ran ∪ ran (comp‘𝑌) ↑_m (Base‘𝑋)) ∈ V
49		ovex 7476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑋 Nat 𝑌) ∈ V
50	49	rnex 7946	. . . . . . . . . . . . . . . . . . 19 ⊢ ran (𝑋 Nat 𝑌) ∈ V
51	50	uniex 7782	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ ran (𝑋 Nat 𝑌) ∈ V
52	51, 51	xpex 7791	. . . . . . . . . . . . . . . . 17 ⊢ (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌)) ∈ V
53		eqid 2763	. . . . . . . . . . . . . . . . . . . . 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 7479	. . . . . . . . . . . . . . . . . . . . . . . 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 7479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥)) ⊆ ∪ ran (comp‘𝑌)
56		rnss 5928	. . . . . . . . . . . . . . . . . . . . . . . . 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 4878	. . . . . . . . . . . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . . . . . . . . . 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 7476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝑌)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥)) ∈ V
61	60	elpw 4564	. . . . . . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . . . . . 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 7136	. . . . . . . . . . . . . . . . . . . 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 6921	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (comp‘𝑌) ∈ V
66	65	rnex 7946	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ran (comp‘𝑌) ∈ V
67	66	uniex 7782	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ ran (comp‘𝑌) ∈ V
68	67	rnex 7946	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ran ∪ ran (comp‘𝑌) ∈ V
69	68	uniex 7782	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∪ ran ∪ ran (comp‘𝑌) ∈ V
70	69	pwex 5346	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ V
71		fvex 6921	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Base‘𝑋) ∈ V
72	70, 71	elmap 8927	. . . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . . 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 3085	. . . . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . . . 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 8104	. . . . . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . . . . 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 7479	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔(𝑋 Nat 𝑌)ℎ) ⊆ ∪ ran (𝑋 Nat 𝑌)
79		ovssunirn 7479	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓(𝑋 Nat 𝑌)𝑔) ⊆ ∪ ran (𝑋 Nat 𝑌)
80		xpss12 5670	. . . . . . . . . . . . . . . . . 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 705	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑔(𝑋 Nat 𝑌)ℎ) × (𝑓(𝑋 Nat 𝑌)𝑔)) ⊆ (∪ ran (𝑋 Nat 𝑌) × ∪ ran (𝑋 Nat 𝑌))
82		elpm2r 8893	. . . . . . . . . . . . . . . . 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 706	. . . . . . . . . . . . . . . 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 3757	. . . . . . . . . . . . . . 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 4377	. . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . 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 3757	. . . . . . . . . . . 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 4377	. . . . . . . . . . . 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 235	. . . . . . . . . . 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 3085	. . . . . . . . 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 2763	. . . . . . . . . 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 8104	. . . . . . . . 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 233	. . . . . . . 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 10839	. . . . . 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 10834	. . . . 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 10823	. . . 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 2864	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝑈)
101	1, 12, 15	fuccat 18177	. . 3 ⊢ (𝜑 → 𝑄 ∈ Cat)
102	100, 101	elind 4100	. 2 ⊢ (𝜑 → 𝑄 ∈ (𝑈 ∩ Cat))
103	102, 10	eleqtrrd 2867	1 ⊢ (𝜑 → 𝑄 ∈ 𝐵)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 ∀wral 3080 Vcvv 3454 [wsbc 3742 ⦋csb 3851 ⊆ wss 3900 ∩ cin 4092 𝒫 cpw 4560 {ctp 4592 ⟨cop 4838 ∪ cuni 4870 ↦ cmpt 5189 × cxp 5653 ran crn 5656 ⟶wf 6554 ‘cfv 6558 (class class class)co 7443 ∈ cmpo 7445 ωcom 7901 1^st c1st 8023 2^nd c2nd 8024 ↑_m cmap 8875 ↑_pm cpm 8876 WUnicwun 10812 ndxcnx 17400 Basecbs 17416 Hom chom 17468 compcco 17469 Catccat 17867 Func cfunc 18058 Nat cnat 18148 FuncCat cfuc 18149 CatCatccatc 18302

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-inf2 9669 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-oadd 8508 df-omul 8509 df-er 8745 df-ec 8747 df-qs 8751 df-map 8877 df-pm 8878 df-ixp 8954 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-wun 10814 df-ni 10984 df-pli 10985 df-mi 10986 df-lti 10987 df-plpq 11020 df-mpq 11021 df-ltpq 11022 df-enq 11023 df-nq 11024 df-erq 11025 df-plq 11026 df-mq 11027 df-1nq 11028 df-rq 11029 df-ltnq 11030 df-np 11093 df-plp 11095 df-ltp 11097 df-enr 11167 df-nr 11168 df-c 11233 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-fz 13667 df-struct 17354 df-slot 17389 df-ndx 17401 df-base 17417 df-hom 17481 df-cco 17482 df-cat 17871 df-cid 17872 df-func 18062 df-nat 18150 df-fuc 18151 df-catc 18303

This theorem is used by: (None)

W3C validator