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Theorem utopreg 24261
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 24243 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
32adantr 480 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → (unifTop‘𝑈) ∈ Top)
41, 3eqeltrid 2845 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
5 simp-4l 783 . . . . . . . . 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 726 . . . . . . . . 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 769 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑤𝑈)
9 simp-4l 783 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
10 simpr 484 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑤𝑈)
114ad3antrrr 730 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝐽 ∈ Top)
12 simpllr 776 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎𝐽)
13 eqid 2737 . . . . . . . . . . . . . 14 𝐽 = 𝐽
1413eltopss 22913 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑎𝐽) → 𝑎 𝐽)
1511, 12, 14syl2anc 584 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎 𝐽)
16 utopbas 24244 . . . . . . . . . . . . . 14 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
171unieqi 4919 . . . . . . . . . . . . . 14 𝐽 = (unifTop‘𝑈)
1816, 17eqtr4di 2795 . . . . . . . . . . . . 13 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = 𝐽)
199, 18syl 17 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑋 = 𝐽)
2015, 19sseqtrrd 4021 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎𝑋)
21 simplr 769 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑥𝑎)
2220, 21sseldd 3984 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑥𝑋)
231utopsnnei 24258 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑥𝑋) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
249, 10, 22, 23syl3anc 1373 . . . . . . . . 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 23116 . . . . . . . 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 771 . . . . . . . . . . 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 3484 . . . . . . . . . . . 12 𝑥 ∈ V
3029snss 4785 . . . . . . . . . . 11 (𝑥𝑏 ↔ {𝑥} ⊆ 𝑏)
3128, 30sylibr 234 . . . . . . . . . 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 726 . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . 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 4021 . . . . . . . . . . . . . . . . 17 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎𝑋)
41 simpr 484 . . . . . . . . . . . . . . . . 17 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑥𝑎)
4240, 41sseldd 3984 . . . . . . . . . . . . . . . 16 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑥𝑋)
4342ad6antr 736 . . . . . . . . . . . . . . 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 24237 . . . . . . . . . . . . . . 15 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑥𝑋) → (𝑤 “ {𝑥}) ⊆ 𝑋)
4535, 36, 43, 44syl3anc 1373 . . . . . . . . . . . . . 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 4020 . . . . . . . . . . . . 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 773 . . . . . . . . . . . . 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 23062 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ⊆ 𝐽𝑏 ⊆ (𝑤 “ {𝑥})) → ((cls‘𝐽)‘𝑏) ⊆ ((cls‘𝐽)‘(𝑤 “ {𝑥})))
5032, 47, 48, 49syl3anc 1373 . . . . . . . . . . . 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 24213 . . . . . . . . . . . . . . . . 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 5717 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (𝑋 × 𝑋) = ( 𝐽 × 𝐽))
5552, 54sseqtrd 4020 . . . . . . . . . . . . . . 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 786 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑥𝑎)
5856, 57sseldd 3984 . . . . . . . . . . . . . . 15 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑥 𝐽)
5913, 13imasncls 23700 . . . . . . . . . . . . . . 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 839 . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . 17 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑤 = 𝑤)
621utop3cls 24260 . . . . . . . . . . . . . . . . 17 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ⊆ (𝑋 × 𝑋)) ∧ (𝑤𝑈𝑤 = 𝑤)) → ((cls‘(𝐽 ×t 𝐽))‘𝑤) ⊆ (𝑤 ∘ (𝑤𝑤)))
6334, 52, 8, 61, 62syl22anc 839 . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)
6563, 64sstrd 3994 . . . . . . . . . . . . . . 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 6119 . . . . . . . . . . . . . . 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 3994 . . . . . . . . . . . . 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 726 . . . . . . . . . . . 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 3994 . . . . . . . . . . 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 786 . . . . . . . . . . 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 4021 . . . . . . . . . 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 511 . . . . . . . . 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 412 . . . . . . . 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 3168 . . . . . . 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 785 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑈 ∈ (UnifOn‘𝑋))
78 simplr 769 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑣𝑈)
79 ustex3sym 24226 . . . . . . 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 3162 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → ∃𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
82 opnneip 23127 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑎𝐽𝑥𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
836, 37, 41, 82syl3anc 1373 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
841utopsnneip 24257 . . . . . . . 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 2843 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})))
87 eqid 2737 . . . . . . . 8 (𝑣𝑈 ↦ (𝑣 “ {𝑥})) = (𝑣𝑈 ↦ (𝑣 “ {𝑥}))
8887elrnmpt 5969 . . . . . . 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 232 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑥}))
9181, 90r19.29a 3162 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ∃𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
9291ralrimiva 3146 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) → ∀𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
9392ralrimiva 3146 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → ∀𝑎𝐽𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
94 isreg 23340 . 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 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951  {csn 4626   cuni 4907  cmpt 5225   × cxp 5683  ccnv 5684  ran crn 5686  cima 5688  ccom 5689  cfv 6561  (class class class)co 7431  Topctop 22899  clsccl 23026  neicnei 23105  Hauscha 23316  Regcreg 23317   ×t ctx 23568  UnifOncust 24208  unifTopcutop 24239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-map 8868  df-en 8986  df-fin 8989  df-fi 9451  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-cn 23235  df-cnp 23236  df-reg 23324  df-tx 23570  df-ust 24209  df-utop 24240
This theorem is referenced by:  uspreg  24283
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