Theorem pt1hmeo 23691

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 5681	. . . . 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 4587	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → {𝑘} = {𝐴})
5	4	xpeq1d 5648	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → ({𝑘} × {𝑥}) = ({𝐴} × {𝑥}))
6		opeq1 4824	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → ⟨𝑘, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
7	6	sneqd 4589	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → {⟨𝑘, 𝑥⟩} = {⟨𝐴, 𝑥⟩})
8	5, 7	eqeq12d 2745	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩} ↔ ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩}))
9		vex 3440	. . . . . . . 8 ⊢ 𝑘 ∈ V
10		vex 3440	. . . . . . . 8 ⊢ 𝑥 ∈ V
11	9, 10	xpsn 7075	. . . . . . 7 ⊢ ({𝑘} × {𝑥}) = {⟨𝑘, 𝑥⟩}
12	8, 11	vtoclg 3509	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
13	3, 12	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ({𝐴} × {𝑥}) = {⟨𝐴, 𝑥⟩})
14	1, 13	eqtr3id 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ {𝐴} ↦ 𝑥) = {⟨𝐴, 𝑥⟩})
15	14	mpteq2dva 5185	. . 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 5375	. . . . 5 ⊢ {𝐴} ∈ V
19	18	a1i 11	. . . 4 ⊢ (𝜑 → {𝐴} ∈ V)
20		topontop 22798	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
21	17, 20	syl 17	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Top)
22	2, 21	fsnd 6807	. . . 4 ⊢ (𝜑 → {⟨𝐴, 𝐽⟩}:{𝐴}⟶Top)
23	17	cnmptid 23546	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
24	23	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn 𝐽))
25		elsni 4594	. . . . . . . 8 ⊢ (𝑘 ∈ {𝐴} → 𝑘 = 𝐴)
26	25	fveq2d 6826	. . . . . . 7 ⊢ (𝑘 ∈ {𝐴} → ({⟨𝐴, 𝐽⟩}‘𝑘) = ({⟨𝐴, 𝐽⟩}‘𝐴))
27		fvsng 7116	. . . . . . . 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 2786	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → ({⟨𝐴, 𝐽⟩}‘𝑘) = 𝐽)
30	29	oveq2d 7365	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)) = (𝐽 Cn 𝐽))
31	24, 30	eleqtrrd 2831	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴}) → (𝑥 ∈ 𝑋 ↦ 𝑥) ∈ (𝐽 Cn ({⟨𝐴, 𝐽⟩}‘𝑘)))
32	16, 17, 19, 22, 31	ptcn 23512	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ {𝐴} ↦ 𝑥)) ∈ (𝐽 Cn 𝐾))
33	15, 32	eqeltrrd 2829	. 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 2767	. . . . . . . 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 7049	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → (𝑘 ∈ {𝐴} ↦ 𝑥):{𝐴}⟶𝑋)
40		toponmax 22811	. . . . . . . . . . . 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 8766	. . . . . . . . . 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 2828	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 = {⟨𝐴, 𝑥⟩})) → 𝑦 ∈ (𝑋 ↑m {𝐴}))
47	34	fveq1d 6824	. . . . . . . 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 7116	. . . . . . . . 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 2765	. . . . . . 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 8766	. . . . . . . . . . 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 4612	. . . . . . . . . . 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 7019	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → (𝑦‘𝐴) ∈ 𝑋)
63	53, 62	eqeltrd 2828	. . . . . . 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 7072	. . . . . . . . . . 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 4831	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → ⟨𝐴, 𝑥⟩ = ⟨𝐴, (𝑦‘𝐴)⟩)
70	69	sneqd 4589	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (𝑋 ↑m {𝐴}) ∧ 𝑥 = (𝑦‘𝐴))) → {⟨𝐴, 𝑥⟩} = {⟨𝐴, (𝑦‘𝐴)⟩})
71	68, 70	eqtr4d 2767	. . . . . . 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 6088	. . . 4 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 ∈ (𝑋 ↑m {𝐴}) ↦ (𝑦‘𝐴)))
75		xpsng 7073	. . . . . . . . . . 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 2735	. . . . . . . . 9 ⊢ (𝜑 → {⟨𝐴, 𝐽⟩} = ({𝐴} × {𝐽}))
78	77	fveq2d 6826	. . . . . . . 8 ⊢ (𝜑 → (∏t‘{⟨𝐴, 𝐽⟩}) = (∏t‘({𝐴} × {𝐽})))
79	16, 78	eqtrid 2776	. . . . . . 7 ⊢ (𝜑 → 𝐾 = (∏t‘({𝐴} × {𝐽})))
80		eqid 2729	. . . . . . . . 9 ⊢ (∏t‘({𝐴} × {𝐽})) = (∏t‘({𝐴} × {𝐽}))
81	80	pttoponconst 23482	. . . . . . . 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 2828	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘(𝑋 ↑m {𝐴})))
84		toponuni 22799	. . . . . 6 ⊢ (𝐾 ∈ (TopOn‘(𝑋 ↑m {𝐴})) → (𝑋 ↑m {𝐴}) = ∪ 𝐾)
85	83, 84	syl 17	. . . . 5 ⊢ (𝜑 → (𝑋 ↑m {𝐴}) = ∪ 𝐾)
86	85	mpteq1d 5182	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (𝑋 ↑m {𝐴}) ↦ (𝑦‘𝐴)) = (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)))
87	74, 86	eqtrd 2764	. . 3 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) = (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)))
88		eqid 2729	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
89	88, 16	ptpjcn 23496	. . . . 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 7365	. . . 4 ⊢ (𝜑 → (𝐾 Cn ({⟨𝐴, 𝐽⟩}‘𝐴)) = (𝐾 Cn 𝐽))
92	90, 91	eleqtrd 2830	. . 3 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝐾 ↦ (𝑦‘𝐴)) ∈ (𝐾 Cn 𝐽))
93	87, 92	eqeltrd 2828	. 2 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋 ↦ {⟨𝐴, 𝑥⟩}) ∈ (𝐾 Cn 𝐽))
94		ishmeo 23644	. 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 3436  {csn 4577  ⟨cop 4583  ∪ cuni 4858   ↦ cmpt 5173   × cxp 5617  ^◡ccnv 5618  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753  ∏_tcpt 17342  Topctop 22778  TopOnctopon 22795   Cn ccn 23109  Homeochmeo 23638

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-1o 8388  df-2o 8389  df-map 8755  df-ixp 8825  df-en 8873  df-dom 8874  df-fin 8876  df-fi 9301  df-topgen 17347  df-pt 17348  df-top 22779  df-topon 22796  df-bases 22831  df-cn 23112  df-cnp 23113  df-hmeo 23640

This theorem is referenced by:  xpstopnlem1  23694  ptcmpfi  23698