Theorem pt1hmeo 23868

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 5711	. . . . 5 ⊢ ({𝐴} × {𝑥}) = (𝑘 ∈ {𝐴} ↦ 𝑥)
2		pt1hmeo.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3	2	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑉)
4		sneq 4594	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → {𝑘} = {𝐴})
5	4	xpeq1d 5678	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → ({𝑘} × {𝑥}) = ({𝐴} × {𝑥}))
6		opeq1 4833	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → ⟨𝑘, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
7	6	sneqd 4596	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → {⟨𝑘, 𝑥⟩} = {⟨𝐴, 𝑥⟩})
8	5, 7	eqeq12d 2780	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩} ↔ ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩}))
9		vex 3460	. . . . . . . 8 ⊢ 𝑘 ∈ V
10		vex 3460	. . . . . . . 8 ⊢ 𝑥 ∈ V
11	9, 10	xpsn 7125	. . . . . . 7 ⊢ ({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩}
12	8, 11	vtoclg 3524	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
13	3, 12	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
14	1, 13	eqtr3id 2813	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
15	14	mpteq2dva 5195	. . 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 5398	. . . . 5 ⊢ {𝐴} ∈ V
19	18	a1i 11	. . . 4 ⊢ (𝜑 → {𝐴} ∈ V)
20		topontop 22975	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21	17, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
22	2, 21	fsnd 6853	. . . 4 ⊢ (𝜑 → {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top)
23	17	cnmptid 23723	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
24	23	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
25		elsni 4601	. . . . . . . 8 ⊢ (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
26	25	fveq2d 6873	. . . . . . 7 ⊢ (𝑘 ∈ {𝐴} → ({⟨𝐴, 𝐽⟩}‘𝑘) = ({⟨𝐴, 𝐽⟩}‘𝐴))
27		fvsng 7166	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
28	2, 17, 27	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → ({⟨𝐴, 𝐽⟩}‘𝐴) = 𝐽)
29	26, 28	sylan9eqr 2821	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → ({⟨𝐴, 𝐽⟩}‘𝑘) = 𝐽)
30	29	oveq2d 7414	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)) = (𝐽 Cn 𝐽))
31	24, 30	eleqtrrd 2867	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)))
32	16, 17, 19, 22, 31	ptcn 23689	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ {𝐴} ↦ 𝑥)) ∈ (𝐽 Cn 𝐾))
33	15, 32	eqeltrrd 2865	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾))
34		simprr 782	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = {⟨𝐴, 𝑥⟩})
35	14	adantrr 727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
36	34, 35	eqtr4d 2802	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = (𝑘 ∈ {𝐴} ↦ 𝑥))
37		simprl 780	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥 ∈ 𝑋)
38	37	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) ∧ 𝑘 ∈ {𝐴}) → 𝑥 ∈ 𝑋)
39	38	fmpttd 7098	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋)
40		toponmax 22988	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
41	17, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
42	41	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑋 ∈ 𝐽)
43		elmapg 8822	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐽 ∧ {𝐴} ∈ V) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋 ↑m {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
44	42, 18, 43	sylancl 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → ((𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋 ↑m {𝐴}) ↔ (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋))
45	39, 44	mpbird 259	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋 ↑m {𝐴}))
46	36, 45	eqeltrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 ∈ (𝑋 ↑m {𝐴}))
47	34	fveq1d 6871	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦‘𝐴) = ({⟨𝐴, 𝑥⟩}‘𝐴))
48	2	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝐴 ∈ 𝑉)
49		fvsng 7166	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
50	48, 37, 49	syl2anc 593	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → ({⟨𝐴, 𝑥⟩}‘𝐴) = 𝑥)
51	47, 50	eqtr2d 2800	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥 = (𝑦‘𝐴))
52	46, 51	jca 519	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴)))
53		simprr 782	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑥 = (𝑦‘𝐴))
54		simprl 780	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑦 ∈ (𝑋 ↑m {𝐴}))
55	41	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑋 ∈ 𝐽)
56		elmapg 8822	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐽 ∧ {𝐴} ∈ V) → (𝑦 ∈ (𝑋 ↑m {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
57	55, 18, 56	sylancl 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → (𝑦 ∈ (𝑋 ↑m {𝐴}) ↔ 𝑦:{𝐴}⟶𝑋))
58	54, 57	mpbid 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑦:{𝐴}⟶𝑋)
59		snidg 4621	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
60	2, 59	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
61	60	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝐴 ∈ {𝐴})
62	58, 61	ffvelcdmd 7068	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → (𝑦‘𝐴) ∈ 𝑋)
63	53, 62	eqeltrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑥 ∈ 𝑋)
64	2	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝐴 ∈ 𝑉)
65		fsn2g 7122	. . . . . . . . . . 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 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → ((𝑦‘𝐴) ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, (𝑦‘𝐴)⟩}))
68	67	simprd 499	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑦 = {⟨𝐴, (𝑦‘𝐴)⟩})
69	53	opeq2d 4840	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → ⟨𝐴, 𝑥⟩ = ⟨𝐴, (𝑦‘𝐴)⟩)
70	69	sneqd 4596	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → {⟨𝐴, 𝑥⟩} = {⟨𝐴, (𝑦‘𝐴)⟩})
71	68, 70	eqtr4d 2802	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑦 = {⟨𝐴, 𝑥⟩})
72	63, 71	jca 519	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩}))
73	52, 72	impbida 810	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩}) ↔ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))))
74	73	mptcnv 6127	. . . 4 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 ∈ (𝑋 ↑m {𝐴}) ↦ (𝑦‘𝐴)))
75		xpsng 7123	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐽 ∈ (TopOn‘𝑋)) → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
76	2, 17, 75	syl2anc 593	. . . . . . . . . 10 ⊢ (𝜑 → ({𝐴} × {𝐽}) = {⟨𝐴, 𝐽⟩})
77	76	eqcomd 2770	. . . . . . . . 9 ⊢ (𝜑 → {⟨𝐴, 𝐽⟩} = ({𝐴} × {𝐽}))
78	77	fveq2d 6873	. . . . . . . 8 ⊢ (𝜑 → (∏t‘{⟨𝐴, 𝐽⟩}) = (∏t‘({𝐴} × {𝐽})))
79	16, 78	eqtrid 2811	. . . . . . 7 ⊢ (𝜑 → 𝐾 = (∏t‘({𝐴} × {𝐽})))
80		eqid 2764	. . . . . . . . 9 ⊢ (∏t‘({𝐴} × {𝐽})) = (∏t‘({𝐴} × {𝐽}))
81	80	pttoponconst 23659	. . . . . . . 8 ⊢ (({𝐴} ∈ V ∧ 𝐽 ∈ (TopOn‘𝑋)) → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋 ↑m {𝐴})))
82	19, 17, 81	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → (∏t‘({𝐴} × {𝐽})) ∈ (TopOn‘(𝑋 ↑m {𝐴})))
83	79, 82	eqeltrd 2864	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘(𝑋 ↑m {𝐴})))
84		toponuni 22976	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘(𝑋 ↑m {𝐴})) → (𝑋 ↑m {𝐴}) = ∪ 𝐾)
85	83, 84	syl 17	. . . . 5 ⊢ (𝜑 → (𝑋 ↑m {𝐴}) = ∪ 𝐾)
86	85	mpteq1d 5192	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝑋 ↑m {𝐴}) ↦ (𝑦‘𝐴)) = (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)))
87	74, 86	eqtrd 2799	. . 3 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)))
88		eqid 2764	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
89	88, 16	ptpjcn 23673	. . . . 5 ⊢ (({𝐴} ∈ V ∧ {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top ∧ 𝐴 ∈ {𝐴}) → (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
90	18, 22, 60, 89	mp3an2i 1489	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
91	28	oveq2d 7414	. . . 4 ⊢ (𝜑 → (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)) = (𝐾 Cn 𝐽))
92	90, 91	eleqtrd 2866	. . 3 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)) ∈ (𝐾 Cn 𝐽))
93	87, 92	eqeltrd 2864	. 2 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽))
94		ishmeo 23821	. 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 592	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽Homeo𝐾))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  Vcvv 3456  {csn 4584  ⟨cop 4590  ∪ cuni 4867   ↦ cmpt 5183   × cxp 5647  ^◡ccnv 5648  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398   ↑_m cmap 8810  ∏_tcpt 17469  Topctop 22955  TopOnctopon 22972   Cn ccn 23286  Homeochmeo 23815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-1o 8439  df-2o 8440  df-map 8812  df-ixp 8882  df-en 8930  df-dom 8931  df-fin 8933  df-fi 9359  df-topgen 17474  df-pt 17475  df-top 22956  df-topon 22973  df-bases 23008  df-cn 23289  df-cnp 23290  df-hmeo 23817

This theorem is referenced by:  xpstopnlem1  23871  ptcmpfi  23875