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Theorem utopreg 21866

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 21848	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
3	2	adantr 480	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → (unifTop‘𝑈) ∈ Top)
4	1, 3	syl5eqel 2692	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
5		simp-4l 802	. . . . . . . . 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 758	. . . . . . . . 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 788	. . . . . . . . 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 802	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
10		simpr 476	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑤 ∈ 𝑈)
11	4	ad3antrrr 762	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝐽 ∈ Top)
12		simpllr 795	. . . . . . . . . . . . 13 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑎 ∈ 𝐽)
13		eqid 2610	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
14	13	eltopss 20537	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑎 ∈ 𝐽) → 𝑎 ⊆ ∪ 𝐽)
15	11, 12, 14	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑎 ⊆ ∪ 𝐽)
16		utopbas 21849	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
17	1	unieqi 4381	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ (unifTop‘𝑈)
18	16, 17	syl6eqr 2662	. . . . . . . . . . . . 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	sseqtr4d 3605	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑎 ⊆ 𝑋)
21		simplr 788	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑥 ∈ 𝑎)
22	20, 21	sseldd 3569	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → 𝑥 ∈ 𝑋)
23	1	utopsnnei 21863	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑋) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
24	9, 10, 22, 23	syl3anc 1318	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑤 ∈ 𝑈) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
25	5, 8, 24	syl2anc 691	. . . . . . . 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 20722	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑏 ∈ 𝐽 ({𝑥} ⊆ 𝑏 ∧ 𝑏 ⊆ (𝑤 “ {𝑥})))
27	7, 25, 26	syl2anc 691	. . . . . . 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 790	. . . . . . . . . . 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 3176	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
30	29	snss 4259	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑏 ↔ {𝑥} ⊆ 𝑏)
31	28, 30	sylibr 223	. . . . . . . . . 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 758	. . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . 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 758	. . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ 𝐽)
38	6, 37, 14	syl2anc 691	. . . . . . . . . . . . . . . . . 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	sseqtr4d 3605	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ⊆ 𝑋)
41		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑎)
42	40, 41	sseldd 3569	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑥 ∈ 𝑋)
43	42	ad6antr 768	. . . . . . . . . . . . . . 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 21842	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ 𝑥 ∈ 𝑋) → (𝑤 “ {𝑥}) ⊆ 𝑋)
45	35, 36, 43, 44	syl3anc 1318	. . . . . . . . . . . . . 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 3604	. . . . . . . . . . . . 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 792	. . . . . . . . . . . . 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 20668	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ⊆ ∪ 𝐽 ∧ 𝑏 ⊆ (𝑤 “ {𝑥})) → ((cls‘𝐽)‘𝑏) ⊆ ((cls‘𝐽)‘(𝑤 “ {𝑥})))
50	32, 47, 48, 49	syl3anc 1318	. . . . . . . . . . . 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 21818	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
52	34, 8, 51	syl2anc 691	. . . . . . . . . . . . . . . 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 5065	. . . . . . . . . . . . . . . 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 3604	. . . . . . . . . . . . . . 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 805	. . . . . . . . . . . . . . . 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 3569	. . . . . . . . . . . . . . 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 21305	. . . . . . . . . . . . . . 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 1319	. . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . . 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 21865	. . . . . . . . . . . . . . . . 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 1319	. . . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . 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 3578	. . . . . . . . . . . . . . 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 5419	. . . . . . . . . . . . . . 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 3578	. . . . . . . . . . . . 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 758	. . . . . . . . . . . 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 3578	. . . . . . . . . . 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 805	. . . . . . . . . . 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	sseqtr4d 3605	. . . . . . . . . 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 553	. . . . . . . . 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 449	. . . . . . . 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 3000	. . . . . . 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 804	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑈 ∈ (UnifOn‘𝑋))
78		simplr 788	. . . . . . 7 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) ∧ 𝑣 ∈ 𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑣 ∈ 𝑈)
79		ustex3sym 21831	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑈 (^◡𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤 ∘ 𝑤)) ⊆ 𝑣))
80	77, 78, 79	syl2anc 691	. . . . . 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 3060	. . . . 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 20733	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑎 ∈ 𝐽 ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
83	6, 37, 41, 82	syl3anc 1318	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
84	1	utopsnneip 21862	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ 𝑋) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})))
85	33, 42, 84	syl2anc 691	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})))
86	83, 85	eleqtrd 2690	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → 𝑎 ∈ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})))
87		eqid 2610	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥})) = (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑥}))
88	87	elrnmpt 5293	. . . . . . 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 221	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → ∃𝑣 ∈ 𝑈 𝑎 = (𝑣 “ {𝑥}))
91	81, 90	r19.29a 3060	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) ∧ 𝑥 ∈ 𝑎) → ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
92	91	ralrimiva 2949	. . 3 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎 ∈ 𝐽) → ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
93	92	ralrimiva 2949	. 2 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → ∀𝑎 ∈ 𝐽 ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
94		isreg 20946	. 2 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝐽 ∀𝑥 ∈ 𝑎 ∃𝑏 ∈ 𝐽 (𝑥 ∈ 𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎)))
95	4, 93, 94	sylanbrc 695	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 {csn 4125 ∪ cuni 4372 ↦ cmpt 4643 × cxp 5036 ^◡ccnv 5037 ran crn 5039 “ cima 5041 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 Topctop 20517 clsccl 20632 neicnei 20711 Hauscha 20922 Regcreg 20923 ×_t ctx 21173 UnifOncust 21813 unifTopcutop 21844

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-fin 7845 df-fi 8200 df-topgen 15927 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-cn 20841 df-cnp 20842 df-reg 20930 df-tx 21175 df-ust 21814 df-utop 21845

This theorem is referenced by: uspreg 21888

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