Theorem pt1hmeo 23700

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 5703	. . . . 5 ⊢ ({𝐴} × {𝑥}) = (𝑘 ∈ {𝐴} ↦ 𝑥)
2		pt1hmeo.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑉)
4		sneq 4602	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → {𝑘} = {𝐴})
5	4	xpeq1d 5670	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → ({𝑘} × {𝑥}) = ({𝐴} × {𝑥}))
6		opeq1 4840	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → ⟨𝑘, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
7	6	sneqd 4604	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → {⟨𝑘, 𝑥⟩} = {⟨𝐴, 𝑥⟩})
8	5, 7	eqeq12d 2746	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩} ↔ ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩}))
9		vex 3454	. . . . . . . 8 ⊢ 𝑘 ∈ V
10		vex 3454	. . . . . . . 8 ⊢ 𝑥 ∈ V
11	9, 10	xpsn 7116	. . . . . . 7 ⊢ ({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩}
12	8, 11	vtoclg 3523	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
13	3, 12	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
14	1, 13	eqtr3id 2779	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
15	14	mpteq2dva 5203	. . 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 5394	. . . . 5 ⊢ {𝐴} ∈ V
19	18	a1i 11	. . . 4 ⊢ (𝜑 → {𝐴} ∈ V)
20		topontop 22807	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21	17, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
22	2, 21	fsnd 6846	. . . 4 ⊢ (𝜑 → {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top)
23	17	cnmptid 23555	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
24	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
25		elsni 4609	. . . . . . . 8 ⊢ (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
26	25	fveq2d 6865	. . . . . . 7 ⊢ (𝑘 ∈ {𝐴} → ({⟨𝐴, 𝐽⟩}‘𝑘) = ({⟨𝐴, 𝐽⟩}‘𝐴))
27		fvsng 7157	. . . . . . . 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 2787	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → ({⟨𝐴, 𝐽⟩}‘𝑘) = 𝐽)
30	29	oveq2d 7406	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)) = (𝐽 Cn 𝐽))
31	24, 30	eleqtrrd 2832	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)))
32	16, 17, 19, 22, 31	ptcn 23521	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ {𝐴} ↦ 𝑥)) ∈ (𝐽 Cn 𝐾))
33	15, 32	eqeltrrd 2830	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐽 Cn 𝐾))
34		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = {⟨𝐴, 𝑥⟩})
35	14	adantrr 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
36	34, 35	eqtr4d 2768	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 = (𝑘 ∈ {𝐴} ↦ 𝑥))
37		simprl 770	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥 ∈ 𝑋)
38	37	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) ∧ 𝑘 ∈ {𝐴}) → 𝑥 ∈ 𝑋)
39	38	fmpttd 7090	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋)
40		toponmax 22820	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
41	17, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
42	41	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑋 ∈ 𝐽)
43		elmapg 8815	. . . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥) ∈ (𝑋 ↑m {𝐴}))
46	36, 45	eqeltrd 2829	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 ∈ (𝑋 ↑m {𝐴}))
47	34	fveq1d 6863	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦‘𝐴) = ({⟨𝐴, 𝑥⟩}‘𝐴))
48	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝐴 ∈ 𝑉)
49		fvsng 7157	. . . . . . . . 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 2766	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑥 = (𝑦‘𝐴))
52	46, 51	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴)))
53		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑥 = (𝑦‘𝐴))
54		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑦 ∈ (𝑋 ↑m {𝐴}))
55	41	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑋 ∈ 𝐽)
56		elmapg 8815	. . . . . . . . . . 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 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑦:{𝐴}⟶𝑋)
59		snidg 4627	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
60	2, 59	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
61	60	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝐴 ∈ {𝐴})
62	58, 61	ffvelcdmd 7060	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → (𝑦‘𝐴) ∈ 𝑋)
63	53, 62	eqeltrd 2829	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑥 ∈ 𝑋)
64	2	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝐴 ∈ 𝑉)
65		fsn2g 7113	. . . . . . . . . . 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 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → ((𝑦‘𝐴) ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, (𝑦‘𝐴)⟩}))
68	67	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑦 = {⟨𝐴, (𝑦‘𝐴)⟩})
69	53	opeq2d 4847	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → ⟨𝐴, 𝑥⟩ = ⟨𝐴, (𝑦‘𝐴)⟩)
70	69	sneqd 4604	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → {⟨𝐴, 𝑥⟩} = {⟨𝐴, (𝑦‘𝐴)⟩})
71	68, 70	eqtr4d 2768	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → 𝑦 = {⟨𝐴, 𝑥⟩})
72	63, 71	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩}))
73	52, 72	impbida 800	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩}) ↔ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))))
74	73	mptcnv 6115	. . . 4 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 ∈ (𝑋 ↑m {𝐴}) ↦ (𝑦‘𝐴)))
75		xpsng 7114	. . . . . . . . . . 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 2736	. . . . . . . . 9 ⊢ (𝜑 → {⟨𝐴, 𝐽⟩} = ({𝐴} × {𝐽}))
78	77	fveq2d 6865	. . . . . . . 8 ⊢ (𝜑 → (∏t‘{⟨𝐴, 𝐽⟩}) = (∏t‘({𝐴} × {𝐽})))
79	16, 78	eqtrid 2777	. . . . . . 7 ⊢ (𝜑 → 𝐾 = (∏t‘({𝐴} × {𝐽})))
80		eqid 2730	. . . . . . . . 9 ⊢ (∏t‘({𝐴} × {𝐽})) = (∏t‘({𝐴} × {𝐽}))
81	80	pttoponconst 23491	. . . . . . . 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 2829	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘(𝑋 ↑m {𝐴})))
84		toponuni 22808	. . . . . 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 2765	. . 3 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)))
88		eqid 2730	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
89	88, 16	ptpjcn 23505	. . . . 5 ⊢ (({𝐴} ∈ V ∧ {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top ∧ 𝐴 ∈ {𝐴}) → (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
90	18, 22, 60, 89	mp3an2i 1468	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)) ∈ (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)))
91	28	oveq2d 7406	. . . 4 ⊢ (𝜑 → (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)) = (𝐾 Cn 𝐽))
92	90, 91	eleqtrd 2831	. . 3 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)) ∈ (𝐾 Cn 𝐽))
93	87, 92	eqeltrd 2829	. 2 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽))
94		ishmeo 23653	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  {csn 4592  ⟨cop 4598  ∪ cuni 4874   ↦ cmpt 5191   × cxp 5639  ^◡ccnv 5640  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802  ∏_tcpt 17408  Topctop 22787  TopOnctopon 22804   Cn ccn 23118  Homeochmeo 23647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-1o 8437  df-2o 8438  df-map 8804  df-ixp 8874  df-en 8922  df-dom 8923  df-fin 8925  df-fi 9369  df-topgen 17413  df-pt 17414  df-top 22788  df-topon 22805  df-bases 22840  df-cn 23121  df-cnp 23122  df-hmeo 23649

This theorem is referenced by:  xpstopnlem1  23703  ptcmpfi  23707