Theorem pt1hmeo 23157

Description: The canonical homeomorphism from a topological product on a singleton to the topology of the factor. (Contributed by Mario Carneiro, 3-Feb-2015.) (Proof shortened by AV, 18-Apr-2021.)

Hypotheses

Ref	Expression
pt1hmeo.j	⊢ 𝐾 = (∏t‘{⟨𝐴, 𝐽⟩})
pt1hmeo.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
pt1hmeo.r	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))

Assertion

Ref	Expression
pt1hmeo	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝐾   𝜑,𝑥   𝑥,𝑋

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem pt1hmeo

Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fconstmpt 5694	. . . . 5 ⊢ ({𝐴} × {𝑥}) = (𝑘 ∈ {𝐴} ↦ 𝑥)
2		pt1hmeo.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑉)
4		sneq 4596	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → {𝑘} = {𝐴})
5	4	xpeq1d 5662	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → ({𝑘} × {𝑥}) = ({𝐴} × {𝑥}))
6		opeq1 4830	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → ⟨𝑘, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
7	6	sneqd 4598	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → {⟨𝑘, 𝑥⟩} = {⟨𝐴, 𝑥⟩})
8	5, 7	eqeq12d 2752	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩} ↔ ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩}))
9		vex 3449	. . . . . . . 8 ⊢ 𝑘 ∈ V
10		vex 3449	. . . . . . . 8 ⊢ 𝑥 ∈ V
11	9, 10	xpsn 7087	. . . . . . 7 ⊢ ({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩}
12	8, 11	vtoclg 3525	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
13	3, 12	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
14	1, 13	eqtr3id 2790	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
15	14	mpteq2dva 5205	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ {𝐴} ↦ 𝑥)) = (𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}))
16		pt1hmeo.j	. . . 4 ⊢ 𝐾 = (∏t‘{⟨𝐴, 𝐽⟩})
17		pt1hmeo.r	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
18		snex 5388	. . . . 5 ⊢ {𝐴} ∈ V
19	18	a1i 11	. . . 4 ⊢ (𝜑 → {𝐴} ∈ V)
20		topontop 22262	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21	17, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
22	2, 21	fsnd 6827	. . . 4 ⊢ (𝜑 → {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top)
23	17	cnmptid 23012	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
24	23	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
25		elsni 4603	. . . . . . . 8 ⊢ (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
26	25	fveq2d 6846	. . . . . . 7 ⊢ (𝑘 ∈ {𝐴} → ({⟨𝐴, 𝐽⟩}‘𝑘) = ({⟨𝐴, 𝐽⟩}‘𝐴))
27		fvsng 7126	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
28	2, 17, 27	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
29	26, 28	sylan9eqr 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → ({⟨𝐴, 𝐽⟩}‘𝑘) = 𝐽)
30	29	oveq2d 7373	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)) = (𝐽 Cn 𝐽))
31	24, 30	eleqtrrd 2841	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)))
32	16, 17, 19, 22, 31	ptcn 22978	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ {𝐴} ↦ 𝑥)) ∈ (𝐽 Cn 𝐾))
33	15, 32	eqeltrrd 2839	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾))
34		simprr 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = {⟨𝐴, 𝑥⟩})
35	14	adantrr 715	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
36	34, 35	eqtr4d 2779	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = (𝑘 ∈ {𝐴} ↦ 𝑥))
37		simprl 769	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥 ∈ 𝑋)
38	37	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) ∧ 𝑘 ∈ {𝐴}) → 𝑥 ∈ 𝑋)
39	38	fmpttd 7063	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋)
40		toponmax 22275	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
41	17, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
42	41	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑋 ∈ 𝐽)
43		elmapg 8778	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐽 ∧ {𝐴} ∈ V) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋 ↑m {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
44	42, 18, 43	sylancl 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋 ↑m {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
45	39, 44	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋 ↑m {𝐴}))
46	36, 45	eqeltrd 2838	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 ∈ (𝑋 ↑m {𝐴}))
47	34	fveq1d 6844	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦‘𝐴) = ({⟨𝐴, 𝑥⟩}‘𝐴))
48	2	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝐴 ∈ 𝑉)
49		fvsng 7126	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
50	48, 37, 49	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
51	47, 50	eqtr2d 2777	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥 = (𝑦‘𝐴))
52	46, 51	jca 512	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴)))
53		simprr 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑥 = (𝑦‘𝐴))
54		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑦 ∈ (𝑋 ↑m {𝐴}))
55	41	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑋 ∈ 𝐽)
56		elmapg 8778	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐽 ∧ {𝐴} ∈ V) → (𝑦 ∈ (𝑋 ↑m {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
57	55, 18, 56	sylancl 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → (𝑦 ∈ (𝑋 ↑m {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
58	54, 57	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑦:{𝐴}⟶𝑋)
59		snidg 4620	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
60	2, 59	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
61	60	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝐴 ∈ {𝐴})
62	58, 61	ffvelcdmd 7036	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → (𝑦‘𝐴) ∈ 𝑋)
63	53, 62	eqeltrd 2838	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑥 ∈ 𝑋)
64	2	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝐴 ∈ 𝑉)
65		fsn2g 7084	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → (𝑦:{𝐴}⟶𝑋 ↔ ((𝑦‘𝐴) ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, (𝑦‘𝐴)⟩})))
66	64, 65	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → (𝑦:{𝐴}⟶𝑋 ↔ ((𝑦‘𝐴) ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, (𝑦‘𝐴)⟩})))
67	58, 66	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → ((𝑦‘𝐴) ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, (𝑦‘𝐴)⟩}))
68	67	simprd 496	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑦 = {⟨𝐴, (𝑦‘𝐴)⟩})
69	53	opeq2d 4837	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → ⟨𝐴, 𝑥⟩ = ⟨𝐴, (𝑦‘𝐴)⟩)
70	69	sneqd 4598	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → {⟨𝐴, 𝑥⟩} = {⟨𝐴, (𝑦‘𝐴)⟩})
71	68, 70	eqtr4d 2779	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑦 = {⟨𝐴, 𝑥⟩})
72	63, 71	jca 512	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩}))
73	52, 72	impbida 799	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩}) ↔ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))))
74	73	mptcnv 6092	. . . 4 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 ∈ (𝑋 ↑m {𝐴}) ↦ (𝑦‘𝐴)))
75		xpsng 7085	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
76	2, 17, 75	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
77	76	eqcomd 2742	. . . . . . . . 9 ⊢ (𝜑 → {⟨𝐴, 𝐽⟩} = ({𝐴} × {𝐽}))
78	77	fveq2d 6846	. . . . . . . 8 ⊢ (𝜑 → (∏t‘{⟨𝐴, 𝐽⟩}) = (∏t‘({𝐴} × {𝐽})))
79	16, 78	eqtrid 2788	. . . . . . 7 ⊢ (𝜑 → 𝐾 = (∏t‘({𝐴} × {𝐽})))
80		eqid 2736	. . . . . . . . 9 ⊢ (∏t‘({𝐴} × {𝐽})) = (∏t‘({𝐴} × {𝐽}))
81	80	pttoponconst 22948	. . . . . . . 8 ⊢ (({𝐴} ∈ V ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋 ↑m {𝐴})))
82	19, 17, 81	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋 ↑m {𝐴})))
83	79, 82	eqeltrd 2838	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘(𝑋 ↑m {𝐴})))
84		toponuni 22263	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘(𝑋 ↑m {𝐴})) → (𝑋 ↑m {𝐴}) = ∪ 𝐾)
85	83, 84	syl 17	. . . . 5 ⊢ (𝜑 → (𝑋 ↑m {𝐴}) = ∪ 𝐾)
86	85	mpteq1d 5200	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝑋 ↑m {𝐴}) ↦ (𝑦‘𝐴)) = (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)))
87	74, 86	eqtrd 2776	. . 3 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)))
88		eqid 2736	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
89	88, 16	ptpjcn 22962	. . . . 5 ⊢ (({𝐴} ∈ V ∧ {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top ∧ 𝐴 ∈ {𝐴}) → (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
90	18, 22, 60, 89	mp3an2i 1466	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
91	28	oveq2d 7373	. . . 4 ⊢ (𝜑 → (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)) = (𝐾 Cn 𝐽))
92	90, 91	eleqtrd 2840	. . 3 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)) ∈ (𝐾 Cn 𝐽))
93	87, 92	eqeltrd 2838	. 2 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽))
94		ishmeo 23110	. 2 ⊢ ((𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾) ↔ ((𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾) ∧ ◡(𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽)))
95	33, 93, 94	sylanbrc 583	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3445  {csn 4586  ⟨cop 4592  ∪ cuni 4865   ↦ cmpt 5188   × cxp 5631  ^◡ccnv 5632  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ↑_m cmap 8765  ∏_tcpt 17320  Topctop 22242  TopOnctopon 22259   Cn ccn 22575  Homeochmeo 23104

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9347  df-topgen 17325  df-pt 17326  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-cnp 22579  df-hmeo 23106

This theorem is referenced by:  xpstopnlem1  23160  ptcmpfi  23164