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Theorem utopreg 22788
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.
StepHypRef Expression
1 utopreg.1 . . 3 𝐽 = (unifTop‘𝑈)
2 utoptop 22770 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
32adantr 481 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → (unifTop‘𝑈) ∈ Top)
41, 3eqeltrid 2914 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
5 simp-4l 779 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎))
64ad2antrr 722 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝐽 ∈ Top)
75, 6syl 17 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝐽 ∈ Top)
8 simplr 765 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑤𝑈)
9 simp-4l 779 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
10 simpr 485 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑤𝑈)
114ad3antrrr 726 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝐽 ∈ Top)
12 simpllr 772 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎𝐽)
13 eqid 2818 . . . . . . . . . . . . . 14 𝐽 = 𝐽
1413eltopss 21443 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑎𝐽) → 𝑎 𝐽)
1511, 12, 14syl2anc 584 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎 𝐽)
16 utopbas 22771 . . . . . . . . . . . . . 14 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
171unieqi 4839 . . . . . . . . . . . . . 14 𝐽 = (unifTop‘𝑈)
1816, 17syl6eqr 2871 . . . . . . . . . . . . 13 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = 𝐽)
199, 18syl 17 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑋 = 𝐽)
2015, 19sseqtrrd 4005 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎𝑋)
21 simplr 765 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑥𝑎)
2220, 21sseldd 3965 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑥𝑋)
231utopsnnei 22785 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑥𝑋) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
249, 10, 22, 23syl3anc 1363 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
255, 8, 24syl2anc 584 . . . . . . . 8 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
26 neii2 21644 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑏𝐽 ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥})))
277, 25, 26syl2anc 584 . . . . . . 7 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → ∃𝑏𝐽 ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥})))
28 simprl 767 . . . . . . . . . . 11 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → {𝑥} ⊆ 𝑏)
29 vex 3495 . . . . . . . . . . . 12 𝑥 ∈ V
3029snss 4710 . . . . . . . . . . 11 (𝑥𝑏 ↔ {𝑥} ⊆ 𝑏)
3128, 30sylibr 235 . . . . . . . . . 10 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑥𝑏)
327ad2antrr 722 . . . . . . . . . . . . 13 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝐽 ∈ Top)
33 simplll 771 . . . . . . . . . . . . . . . . 17 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑈 ∈ (UnifOn‘𝑋))
345, 33syl 17 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑈 ∈ (UnifOn‘𝑋))
3534ad2antrr 722 . . . . . . . . . . . . . . 15 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑈 ∈ (UnifOn‘𝑋))
368ad2antrr 722 . . . . . . . . . . . . . . 15 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑤𝑈)
37 simplr 765 . . . . . . . . . . . . . . . . . . 19 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎𝐽)
386, 37, 14syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎 𝐽)
3933, 18syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑋 = 𝐽)
4038, 39sseqtrrd 4005 . . . . . . . . . . . . . . . . 17 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎𝑋)
41 simpr 485 . . . . . . . . . . . . . . . . 17 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑥𝑎)
4240, 41sseldd 3965 . . . . . . . . . . . . . . . 16 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑥𝑋)
4342ad6antr 732 . . . . . . . . . . . . . . 15 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑥𝑋)
44 ustimasn 22764 . . . . . . . . . . . . . . 15 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑥𝑋) → (𝑤 “ {𝑥}) ⊆ 𝑋)
4535, 36, 43, 44syl3anc 1363 . . . . . . . . . . . . . 14 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → (𝑤 “ {𝑥}) ⊆ 𝑋)
4635, 18syl 17 . . . . . . . . . . . . . 14 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑋 = 𝐽)
4745, 46sseqtrd 4004 . . . . . . . . . . . . 13 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → (𝑤 “ {𝑥}) ⊆ 𝐽)
48 simprr 769 . . . . . . . . . . . . 13 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑏 ⊆ (𝑤 “ {𝑥}))
4913clsss 21590 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ⊆ 𝐽𝑏 ⊆ (𝑤 “ {𝑥})) → ((cls‘𝐽)‘𝑏) ⊆ ((cls‘𝐽)‘(𝑤 “ {𝑥})))
5032, 47, 48, 49syl3anc 1363 . . . . . . . . . . . 12 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → ((cls‘𝐽)‘𝑏) ⊆ ((cls‘𝐽)‘(𝑤 “ {𝑥})))
51 ustssxp 22740 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
5234, 8, 51syl2anc 584 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑤 ⊆ (𝑋 × 𝑋))
5334, 18syl 17 . . . . . . . . . . . . . . . . 17 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑋 = 𝐽)
5453sqxpeqd 5580 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (𝑋 × 𝑋) = ( 𝐽 × 𝐽))
5552, 54sseqtrd 4004 . . . . . . . . . . . . . . 15 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑤 ⊆ ( 𝐽 × 𝐽))
565, 38syl 17 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑎 𝐽)
57 simp-5r 782 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑥𝑎)
5856, 57sseldd 3965 . . . . . . . . . . . . . . 15 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑥 𝐽)
5913, 13imasncls 22228 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝐽 ∈ Top) ∧ (𝑤 ⊆ ( 𝐽 × 𝐽) ∧ 𝑥 𝐽)) → ((cls‘𝐽)‘(𝑤 “ {𝑥})) ⊆ (((cls‘(𝐽 ×t 𝐽))‘𝑤) “ {𝑥}))
607, 7, 55, 58, 59syl22anc 834 . . . . . . . . . . . . . 14 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → ((cls‘𝐽)‘(𝑤 “ {𝑥})) ⊆ (((cls‘(𝐽 ×t 𝐽))‘𝑤) “ {𝑥}))
61 simprl 767 . . . . . . . . . . . . . . . . 17 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑤 = 𝑤)
621utop3cls 22787 . . . . . . . . . . . . . . . . 17 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ⊆ (𝑋 × 𝑋)) ∧ (𝑤𝑈𝑤 = 𝑤)) → ((cls‘(𝐽 ×t 𝐽))‘𝑤) ⊆ (𝑤 ∘ (𝑤𝑤)))
6334, 52, 8, 61, 62syl22anc 834 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → ((cls‘(𝐽 ×t 𝐽))‘𝑤) ⊆ (𝑤 ∘ (𝑤𝑤)))
64 simprr 769 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)
6563, 64sstrd 3974 . . . . . . . . . . . . . . 15 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → ((cls‘(𝐽 ×t 𝐽))‘𝑤) ⊆ 𝑣)
66 imass1 5957 . . . . . . . . . . . . . . 15 (((cls‘(𝐽 ×t 𝐽))‘𝑤) ⊆ 𝑣 → (((cls‘(𝐽 ×t 𝐽))‘𝑤) “ {𝑥}) ⊆ (𝑣 “ {𝑥}))
6765, 66syl 17 . . . . . . . . . . . . . 14 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (((cls‘(𝐽 ×t 𝐽))‘𝑤) “ {𝑥}) ⊆ (𝑣 “ {𝑥}))
6860, 67sstrd 3974 . . . . . . . . . . . . 13 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → ((cls‘𝐽)‘(𝑤 “ {𝑥})) ⊆ (𝑣 “ {𝑥}))
6968ad2antrr 722 . . . . . . . . . . . 12 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → ((cls‘𝐽)‘(𝑤 “ {𝑥})) ⊆ (𝑣 “ {𝑥}))
7050, 69sstrd 3974 . . . . . . . . . . 11 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → ((cls‘𝐽)‘𝑏) ⊆ (𝑣 “ {𝑥}))
71 simp-5r 782 . . . . . . . . . . 11 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → 𝑎 = (𝑣 “ {𝑥}))
7270, 71sseqtrrd 4005 . . . . . . . . . 10 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → ((cls‘𝐽)‘𝑏) ⊆ 𝑎)
7331, 72jca 512 . . . . . . . . 9 ((((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) ∧ ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥}))) → (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
7473ex 413 . . . . . . . 8 (((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) ∧ 𝑏𝐽) → (({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥})) → (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎)))
7574reximdva 3271 . . . . . . 7 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (∃𝑏𝐽 ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥})) → ∃𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎)))
7627, 75mpd 15 . . . . . 6 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → ∃𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
77 simp-5l 781 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑈 ∈ (UnifOn‘𝑋))
78 simplr 765 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑣𝑈)
79 ustex3sym 22753 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣))
8077, 78, 79syl2anc 584 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣))
8176, 80r19.29a 3286 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → ∃𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
82 opnneip 21655 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑎𝐽𝑥𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
836, 37, 41, 82syl3anc 1363 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
841utopsnneip 22784 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑋) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})))
8533, 42, 84syl2anc 584 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})))
8683, 85eleqtrd 2912 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})))
87 eqid 2818 . . . . . . . 8 (𝑣𝑈 ↦ (𝑣 “ {𝑥})) = (𝑣𝑈 ↦ (𝑣 “ {𝑥}))
8887elrnmpt 5821 . . . . . . 7 (𝑎𝐽 → (𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑥})))
8937, 88syl 17 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → (𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})) ↔ ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑥})))
9086, 89mpbid 233 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑥}))
9181, 90r19.29a 3286 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ∃𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
9291ralrimiva 3179 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) → ∀𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
9392ralrimiva 3179 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → ∀𝑎𝐽𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
94 isreg 21868 . 2 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑎𝐽𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎)))
954, 93, 94sylanbrc 583 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933  {csn 4557   cuni 4830  cmpt 5137   × cxp 5546  ccnv 5547  ran crn 5549  cima 5551  ccom 5552  cfv 6348  (class class class)co 7145  Topctop 21429  clsccl 21554  neicnei 21633  Hauscha 21844  Regcreg 21845   ×t ctx 22096  UnifOncust 22735  unifTopcutop 22766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-fin 8501  df-fi 8863  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-cn 21763  df-cnp 21764  df-reg 21852  df-tx 22098  df-ust 22736  df-utop 22767
This theorem is referenced by:  uspreg  22810
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