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Theorem utopreg 23312

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 23294	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
3	2	adantr 480	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → (unifTop‘𝑈) ∈ Top)
4	1, 3	eqeltrid 2843	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
5		simp-4l 779	. . . . . . . . 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 722	. . . . . . . . 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 765	. . . . . . . . 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 779	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
10		simpr 484	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑤 ∈ 𝑈)
11	4	ad3antrrr 726	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝐽 ∈ Top)
12		simpllr 772	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑎 ∈ 𝐽)
13		eqid 2738	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
14	13	eltopss 21964	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑎 ∈ 𝐽) → 𝑎 ⊆ ∪ 𝐽)
15	11, 12, 14	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑎 ⊆ ∪ 𝐽)
16		utopbas 23295	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
17	1	unieqi 4849	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ (unifTop‘𝑈)
18	16, 17	eqtr4di 2797	. . . . . . . . . . . . 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 3958	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑎 ⊆ 𝑋)
21		simplr 765	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑥 ∈ 𝑎)
22	20, 21	sseldd 3918	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑥 ∈ 𝑋)
23	1	utopsnnei 23309	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑋) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
24	9, 10, 22, 23	syl3anc 1369	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
25	5, 8, 24	syl2anc 583	. . . . . . . 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 22167	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑏 ∈ 𝐽 ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥})))
27	7, 25, 26	syl2anc 583	. . . . . . 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 767	. . . . . . . . . . 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 3426	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
30	29	snss 4716	. . . . . . . . . . 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 722	. . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . 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 765	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ 𝐽)
38	6, 37, 14	syl2anc 583	. . . . . . . . . . . . . . . . . 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 3958	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ⊆ 𝑋)
41		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑎)
42	40, 41	sseldd 3918	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑋)
43	42	ad6antr 732	. . . . . . . . . . . . . . 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 23288	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑋) → (𝑤 “ {𝑥}) ⊆ 𝑋)
45	35, 36, 43, 44	syl3anc 1369	. . . . . . . . . . . . . 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 3957	. . . . . . . . . . . . 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 769	. . . . . . . . . . . . 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 22113	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ⊆ ∪ 𝐽 ∧ 𝑏 ⊆ (𝑤 “ {𝑥})) → ((cls‘𝐽)‘𝑏) ⊆ ((cls‘𝐽)‘(𝑤 “ {𝑥})))
50	32, 47, 48, 49	syl3anc 1369	. . . . . . . . . . . 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 23264	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
52	34, 8, 51	syl2anc 583	. . . . . . . . . . . . . . . 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 5612	. . . . . . . . . . . . . . . 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 3957	. . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . 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 22751	. . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . . 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 23311	. . . . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . 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 3927	. . . . . . . . . . . . . . 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 5998	. . . . . . . . . . . . . . 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 3927	. . . . . . . . . . . . 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 722	. . . . . . . . . . . 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 3927	. . . . . . . . . . 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 782	. . . . . . . . . . 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 3958	. . . . . . . . . 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 511	. . . . . . . . 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 412	. . . . . . . 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 3202	. . . . . . 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 781	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑈 ∈ (UnifOn‘𝑋))
78		simplr 765	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑣 ∈ 𝑈)
79		ustex3sym 23277	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣))
80	77, 78, 79	syl2anc 583	. . . . . 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 3217	. . . . 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 22178	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
83	6, 37, 41, 82	syl3anc 1369	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
84	1	utopsnneip 23308	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})))
85	33, 42, 84	syl2anc 583	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})))
86	83, 85	eleqtrd 2841	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})))
87		eqid 2738	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥}))
88	87	elrnmpt 5854	. . . . . . 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 3217	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
92	91	ralrimiva 3107	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) → ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
93	92	ralrimiva 3107	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → ∀𝑎 ∈ 𝐽 ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
94		isreg 22391	. 2 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝐽 ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎)))
95	4, 93, 94	sylanbrc 582	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 ⊆ wss 3883 {csn 4558 ∪ cuni 4836 ↦ cmpt 5153 × cxp 5578 ^◡ccnv 5579 ran crn 5581 “ cima 5583 ∘ ccom 5584 ‘cfv 6418 (class class class)co 7255 Topctop 21950 clsccl 22077 neicnei 22156 Hauscha 22367 Regcreg 22368 ×_t ctx 22619 UnifOncust 23259 unifTopcutop 23290

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-fin 8695 df-fi 9100 df-topgen 17071 df-top 21951 df-topon 21968 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-cn 22286 df-cnp 22287 df-reg 22375 df-tx 22621 df-ust 23260 df-utop 23291

This theorem is referenced by: uspreg 23334

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