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Theorem utopreg 22858

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 22840	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
3	2	adantr 484	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → (unifTop‘𝑈) ∈ Top)
4	1, 3	eqeltrid 2894	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
5		simp-4l 782	. . . . . . . . 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 725	. . . . . . . . 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 768	. . . . . . . . 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 782	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
10		simpr 488	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑤 ∈ 𝑈)
11	4	ad3antrrr 729	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝐽 ∈ Top)
12		simpllr 775	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑎 ∈ 𝐽)
13		eqid 2798	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
14	13	eltopss 21512	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑎 ∈ 𝐽) → 𝑎 ⊆ ∪ 𝐽)
15	11, 12, 14	syl2anc 587	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑎 ⊆ ∪ 𝐽)
16		utopbas 22841	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
17	1	unieqi 4813	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ (unifTop‘𝑈)
18	16, 17	eqtr4di 2851	. . . . . . . . . . . . 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 3956	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑎 ⊆ 𝑋)
21		simplr 768	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑥 ∈ 𝑎)
22	20, 21	sseldd 3916	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑥 ∈ 𝑋)
23	1	utopsnnei 22855	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑋) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
24	9, 10, 22, 23	syl3anc 1368	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
25	5, 8, 24	syl2anc 587	. . . . . . . 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 21713	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑏 ∈ 𝐽 ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥})))
27	7, 25, 26	syl2anc 587	. . . . . . 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 770	. . . . . . . . . . 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 3444	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
30	29	snss 4679	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑏 ↔ {𝑥} ⊆ 𝑏)
31	28, 30	sylibr 237	. . . . . . . . . 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 725	. . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ 𝐽)
38	6, 37, 14	syl2anc 587	. . . . . . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ⊆ 𝑋)
41		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑎)
42	40, 41	sseldd 3916	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑋)
43	42	ad6antr 735	. . . . . . . . . . . . . . 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 22834	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑋) → (𝑤 “ {𝑥}) ⊆ 𝑋)
45	35, 36, 43, 44	syl3anc 1368	. . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . 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 772	. . . . . . . . . . . . 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 21659	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ⊆ ∪ 𝐽 ∧ 𝑏 ⊆ (𝑤 “ {𝑥})) → ((cls‘𝐽)‘𝑏) ⊆ ((cls‘𝐽)‘(𝑤 “ {𝑥})))
50	32, 47, 48, 49	syl3anc 1368	. . . . . . . . . . . 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 22810	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
52	34, 8, 51	syl2anc 587	. . . . . . . . . . . . . . . 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 5551	. . . . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . 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 3916	. . . . . . . . . . . . . . 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 22297	. . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . 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 22857	. . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . 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 3925	. . . . . . . . . . . . . . 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 5931	. . . . . . . . . . . . . . 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 3925	. . . . . . . . . . . . 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 725	. . . . . . . . . . . 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 3925	. . . . . . . . . . 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 785	. . . . . . . . . . 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 3956	. . . . . . . . . 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 515	. . . . . . . . 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 416	. . . . . . . 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 3233	. . . . . . 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 784	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑈 ∈ (UnifOn‘𝑋))
78		simplr 768	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑣 ∈ 𝑈)
79		ustex3sym 22823	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣))
80	77, 78, 79	syl2anc 587	. . . . . 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 3248	. . . . 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 21724	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
83	6, 37, 41, 82	syl3anc 1368	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
84	1	utopsnneip 22854	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})))
85	33, 42, 84	syl2anc 587	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})))
86	83, 85	eleqtrd 2892	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})))
87		eqid 2798	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥}))
88	87	elrnmpt 5792	. . . . . . 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 235	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑥}))
91	81, 90	r19.29a 3248	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
92	91	ralrimiva 3149	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) → ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
93	92	ralrimiva 3149	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → ∀𝑎 ∈ 𝐽 ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
94		isreg 21937	. 2 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝐽 ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎)))
95	4, 93, 94	sylanbrc 586	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ⊆ wss 3881 {csn 4525 ∪ cuni 4800 ↦ cmpt 5110 × cxp 5517 ^◡ccnv 5518 ran crn 5520 “ cima 5522 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 Topctop 21498 clsccl 21623 neicnei 21702 Hauscha 21913 Regcreg 21914 ×_t ctx 22165 UnifOncust 22805 unifTopcutop 22836

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-fin 8496 df-fi 8859 df-topgen 16709 df-top 21499 df-topon 21516 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-cn 21832 df-cnp 21833 df-reg 21921 df-tx 22167 df-ust 22806 df-utop 22837

This theorem is referenced by: uspreg 22880

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