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Theorem utopreg 23748

Description: All Hausdorff uniform spaces are regular. Proposition 3 of [BourbakiTop1] p. II.5. (Contributed by Thierry Arnoux, 16-Jan-2018.)

**Hypothesis**
Ref	Expression
utopreg.1	⊢ 𝐽 = (unifTop‘𝑈)

**Assertion**
Ref	Expression
utopreg	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg)

Proof of Theorem utopreg Dummy variables 𝑎 𝑏 𝑣 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		utopreg.1	. . 3 ⊢ 𝐽 = (unifTop‘𝑈)
2		utoptop 23730	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
3	2	adantr 481	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → (unifTop‘𝑈) ∈ Top)
4	1, 3	eqeltrid 2837	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
5		simp-4l 781	. . . . . . . . 9 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎))
6	4	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝐽 ∈ Top)
7	5, 6	syl 17	. . . . . . . 8 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → 𝐽 ∈ Top)
8		simplr 767	. . . . . . . . 9 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → 𝑤 ∈ 𝑈)
9		simp-4l 781	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
10		simpr 485	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑤 ∈ 𝑈)
11	4	ad3antrrr 728	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝐽 ∈ Top)
12		simpllr 774	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑎 ∈ 𝐽)
13		eqid 2732	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
14	13	eltopss 22400	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑎 ∈ 𝐽) → 𝑎 ⊆ ∪ 𝐽)
15	11, 12, 14	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑎 ⊆ ∪ 𝐽)
16		utopbas 23731	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
17	1	unieqi 4920	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ (unifTop‘𝑈)
18	16, 17	eqtr4di 2790	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ 𝐽)
19	9, 18	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑋 = ∪ 𝐽)
20	15, 19	sseqtrrd 4022	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑎 ⊆ 𝑋)
21		simplr 767	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑥 ∈ 𝑎)
22	20, 21	sseldd 3982	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑥 ∈ 𝑋)
23	1	utopsnnei 23745	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑋) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
24	9, 10, 22, 23	syl3anc 1371	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
25	5, 8, 24	syl2anc 584	. . . . . . . 8 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
26		neii2 22603	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑏 ∈ 𝐽 ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥})))
27	7, 25, 26	syl2anc 584	. . . . . . 7 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → ∃𝑏 ∈ 𝐽 ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥})))
28		simprl 769	. . . . . . . . . . 11 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → {𝑥} ⊆ 𝑏)
29		vex 3478	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
30	29	snss 4788	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑏 ↔ {𝑥} ⊆ 𝑏)
31	28, 30	sylibr 233	. . . . . . . . . 10 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑥 ∈ 𝑏)
32	7	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝐽 ∈ Top)
33		simplll 773	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
34	5, 33	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → 𝑈 ∈ (UnifOn‘𝑋))
35	34	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑈 ∈ (UnifOn‘𝑋))
36	8	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑤 ∈ 𝑈)
37		simplr 767	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ 𝐽)
38	6, 37, 14	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ⊆ ∪ 𝐽)
39	33, 18	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑋 = ∪ 𝐽)
40	38, 39	sseqtrrd 4022	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ⊆ 𝑋)
41		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑎)
42	40, 41	sseldd 3982	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑋)
43	42	ad6antr 734	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑥 ∈ 𝑋)
44		ustimasn 23724	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑋) → (𝑤 “ {𝑥}) ⊆ 𝑋)
45	35, 36, 43, 44	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → (𝑤 “ {𝑥}) ⊆ 𝑋)
46	35, 18	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑋 = ∪ 𝐽)
47	45, 46	sseqtrd 4021	. . . . . . . . . . . . 13 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → (𝑤 “ {𝑥}) ⊆ ∪ 𝐽)
48		simprr 771	. . . . . . . . . . . . 13 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑏 ⊆ (𝑤 “ {𝑥}))
49	13	clsss 22549	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ⊆ ∪ 𝐽 ∧ 𝑏 ⊆ (𝑤 “ {𝑥})) → ((cls‘𝐽)‘𝑏) ⊆ ((cls‘𝐽)‘(𝑤 “ {𝑥})))
50	32, 47, 48, 49	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → ((cls‘𝐽)‘𝑏) ⊆ ((cls‘𝐽)‘(𝑤 “ {𝑥})))
51		ustssxp 23700	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
52	34, 8, 51	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → 𝑤 ⊆ (𝑋 × 𝑋))
53	34, 18	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → 𝑋 = ∪ 𝐽)
54	53	sqxpeqd 5707	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → (𝑋 × 𝑋) = (∪ 𝐽 × ∪ 𝐽))
55	52, 54	sseqtrd 4021	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → 𝑤 ⊆ (∪ 𝐽 × ∪ 𝐽))
56	5, 38	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → 𝑎 ⊆ ∪ 𝐽)
57		simp-5r 784	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → 𝑥 ∈ 𝑎)
58	56, 57	sseldd 3982	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → 𝑥 ∈ ∪ 𝐽)
59	13, 13	imasncls 23187	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝐽 ∈ Top) ∧ (𝑤 ⊆ (∪ 𝐽 × ∪ 𝐽) ∧ 𝑥 ∈ ∪ 𝐽)) → ((cls‘𝐽)‘(𝑤 “ {𝑥})) ⊆ (((cls‘(𝐽 ×_t 𝐽))‘𝑤) “ {𝑥}))
60	7, 7, 55, 58, 59	syl22anc 837	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → ((cls‘𝐽)‘(𝑤 “ {𝑥})) ⊆ (((cls‘(𝐽 ×_t 𝐽))‘𝑤) “ {𝑥}))
61		simprl 769	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → ^◡𝑤 = 𝑤)
62	1	utop3cls 23747	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ⊆ (𝑋 × 𝑋)) ∧ (𝑤 ∈ 𝑈 ∧ ^◡𝑤 = 𝑤)) → ((cls‘(𝐽 ×_t 𝐽))‘𝑤) ⊆ (𝑤 ∘ (𝑤 ∘ 𝑤)))
63	34, 52, 8, 61, 62	syl22anc 837	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → ((cls‘(𝐽 ×_t 𝐽))‘𝑤) ⊆ (𝑤 ∘ (𝑤 ∘ 𝑤)))
64		simprr 771	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)
65	63, 64	sstrd 3991	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → ((cls‘(𝐽 ×_t 𝐽))‘𝑤) ⊆ 𝑣)
66		imass1 6097	. . . . . . . . . . . . . . 15 ⊢ (((cls‘(𝐽 ×_t 𝐽))‘𝑤) ⊆ 𝑣 → (((cls‘(𝐽 ×_t 𝐽))‘𝑤) “ {𝑥}) ⊆ (𝑣 “ {𝑥}))
67	65, 66	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → (((cls‘(𝐽 ×_t 𝐽))‘𝑤) “ {𝑥}) ⊆ (𝑣 “ {𝑥}))
68	60, 67	sstrd 3991	. . . . . . . . . . . . 13 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → ((cls‘𝐽)‘(𝑤 “ {𝑥})) ⊆ (𝑣 “ {𝑥}))
69	68	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → ((cls‘𝐽)‘(𝑤 “ {𝑥})) ⊆ (𝑣 “ {𝑥}))
70	50, 69	sstrd 3991	. . . . . . . . . . 11 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → ((cls‘𝐽)‘𝑏) ⊆ (𝑣 “ {𝑥}))
71		simp-5r 784	. . . . . . . . . . 11 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑎 = (𝑣 “ {𝑥}))
72	70, 71	sseqtrrd 4022	. . . . . . . . . 10 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → ((cls‘𝐽)‘𝑏) ⊆ 𝑎)
73	31, 72	jca 512	. . . . . . . . 9 ⊢ ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) ∧ ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥}))) → (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
74	73	ex 413	. . . . . . . 8 ⊢ (((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) ∧ 𝑏 ∈ 𝐽) → (({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥})) → (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎)))
75	74	reximdva 3168	. . . . . . 7 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → (∃𝑏 ∈ 𝐽 ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥})) → ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎)))
76	27, 75	mpd 15	. . . . . 6 ⊢ ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤 ∈ 𝑈) ∧ (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣)) → ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
77		simp-5l 783	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑈 ∈ (UnifOn‘𝑋))
78		simplr 767	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑣 ∈ 𝑈)
79		ustex3sym 23713	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣))
80	77, 78, 79	syl2anc 584	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → ∃𝑤 ∈ 𝑈 (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣))
81	76, 80	r19.29a 3162	. . . . 5 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
82		opnneip 22614	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
83	6, 37, 41, 82	syl3anc 1371	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
84	1	utopsnneip 23744	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})))
85	33, 42, 84	syl2anc 584	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})))
86	83, 85	eleqtrd 2835	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})))
87		eqid 2732	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥}))
88	87	elrnmpt 5953	. . . . . . 7 ⊢ (𝑎 ∈ 𝐽 → (𝑎 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})) ↔ ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑥})))
89	37, 88	syl 17	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → (𝑎 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})) ↔ ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑥})))
90	86, 89	mpbid 231	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑥}))
91	81, 90	r19.29a 3162	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
92	91	ralrimiva 3146	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) → ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
93	92	ralrimiva 3146	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → ∀𝑎 ∈ 𝐽 ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
94		isreg 22827	. 2 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝐽 ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎)))
95	4, 93, 94	sylanbrc 583	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ⊆ wss 3947 {csn 4627 ∪ cuni 4907 ↦ cmpt 5230 × cxp 5673 ^◡ccnv 5674 ran crn 5676 “ cima 5678 ∘ ccom 5679 ‘cfv 6540 (class class class)co 7405 Topctop 22386 clsccl 22513 neicnei 22592 Hauscha 22803 Regcreg 22804 ×_t ctx 23055 UnifOncust 23695 unifTopcutop 23726

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-fin 8939 df-fi 9402 df-topgen 17385 df-top 22387 df-topon 22404 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-cn 22722 df-cnp 22723 df-reg 22811 df-tx 23057 df-ust 23696 df-utop 23727

This theorem is referenced by: uspreg 23770

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