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Theorem utopreg 22837
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 22819 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
32adantr 483 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → (unifTop‘𝑈) ∈ Top)
41, 3eqeltrid 2915 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
5 simp-4l 781 . . . . . . . . 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 724 . . . . . . . . 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 767 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑤𝑈)
9 simp-4l 781 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
10 simpr 487 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑤𝑈)
114ad3antrrr 728 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝐽 ∈ Top)
12 simpllr 774 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎𝐽)
13 eqid 2820 . . . . . . . . . . . . . 14 𝐽 = 𝐽
1413eltopss 21491 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑎𝐽) → 𝑎 𝐽)
1511, 12, 14syl2anc 586 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎 𝐽)
16 utopbas 22820 . . . . . . . . . . . . . 14 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
171unieqi 4827 . . . . . . . . . . . . . 14 𝐽 = (unifTop‘𝑈)
1816, 17syl6eqr 2873 . . . . . . . . . . . . 13 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = 𝐽)
199, 18syl 17 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑋 = 𝐽)
2015, 19sseqtrrd 3987 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎𝑋)
21 simplr 767 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑥𝑎)
2220, 21sseldd 3947 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑥𝑋)
231utopsnnei 22834 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑥𝑋) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
249, 10, 22, 23syl3anc 1367 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
255, 8, 24syl2anc 586 . . . . . . . 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 21692 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑏𝐽 ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥})))
277, 25, 26syl2anc 586 . . . . . . 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 769 . . . . . . . . . . 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 3476 . . . . . . . . . . . 12 𝑥 ∈ V
3029snss 4694 . . . . . . . . . . 11 (𝑥𝑏 ↔ {𝑥} ⊆ 𝑏)
3128, 30sylibr 236 . . . . . . . . . 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 724 . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . 19 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎𝐽)
386, 37, 14syl2anc 586 . . . . . . . . . . . . . . . . . 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 3987 . . . . . . . . . . . . . . . . 17 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎𝑋)
41 simpr 487 . . . . . . . . . . . . . . . . 17 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑥𝑎)
4240, 41sseldd 3947 . . . . . . . . . . . . . . . 16 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑥𝑋)
4342ad6antr 734 . . . . . . . . . . . . . . 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 22813 . . . . . . . . . . . . . . 15 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑥𝑋) → (𝑤 “ {𝑥}) ⊆ 𝑋)
4535, 36, 43, 44syl3anc 1367 . . . . . . . . . . . . . 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 3986 . . . . . . . . . . . . 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 771 . . . . . . . . . . . . 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 21638 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ⊆ 𝐽𝑏 ⊆ (𝑤 “ {𝑥})) → ((cls‘𝐽)‘𝑏) ⊆ ((cls‘𝐽)‘(𝑤 “ {𝑥})))
5032, 47, 48, 49syl3anc 1367 . . . . . . . . . . . 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 22789 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
5234, 8, 51syl2anc 586 . . . . . . . . . . . . . . . 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 5563 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (𝑋 × 𝑋) = ( 𝐽 × 𝐽))
5552, 54sseqtrd 3986 . . . . . . . . . . . . . . 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 784 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑥𝑎)
5856, 57sseldd 3947 . . . . . . . . . . . . . . 15 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑥 𝐽)
5913, 13imasncls 22276 . . . . . . . . . . . . . . 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 836 . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . 17 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑤 = 𝑤)
621utop3cls 22836 . . . . . . . . . . . . . . . . 17 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ⊆ (𝑋 × 𝑋)) ∧ (𝑤𝑈𝑤 = 𝑤)) → ((cls‘(𝐽 ×t 𝐽))‘𝑤) ⊆ (𝑤 ∘ (𝑤𝑤)))
6334, 52, 8, 61, 62syl22anc 836 . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)
6563, 64sstrd 3956 . . . . . . . . . . . . . . 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 5940 . . . . . . . . . . . . . . 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 3956 . . . . . . . . . . . . 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 724 . . . . . . . . . . . 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 3956 . . . . . . . . . . 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 784 . . . . . . . . . . 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 3987 . . . . . . . . . 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 514 . . . . . . . . 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 415 . . . . . . . 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 3261 . . . . . . 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 783 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑈 ∈ (UnifOn‘𝑋))
78 simplr 767 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑣𝑈)
79 ustex3sym 22802 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣))
8077, 78, 79syl2anc 586 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣))
8176, 80r19.29a 3276 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → ∃𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
82 opnneip 21703 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑎𝐽𝑥𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
836, 37, 41, 82syl3anc 1367 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
841utopsnneip 22833 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑋) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})))
8533, 42, 84syl2anc 586 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})))
8683, 85eleqtrd 2913 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})))
87 eqid 2820 . . . . . . . 8 (𝑣𝑈 ↦ (𝑣 “ {𝑥})) = (𝑣𝑈 ↦ (𝑣 “ {𝑥}))
8887elrnmpt 5804 . . . . . . 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 234 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑥}))
9181, 90r19.29a 3276 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ∃𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
9291ralrimiva 3169 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) → ∀𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
9392ralrimiva 3169 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → ∀𝑎𝐽𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
94 isreg 21916 . 2 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑎𝐽𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎)))
954, 93, 94sylanbrc 585 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  wral 3125  wrex 3126  wss 3913  {csn 4543   cuni 4814  cmpt 5122   × cxp 5529  ccnv 5530  ran crn 5532  cima 5534  ccom 5535  cfv 6331  (class class class)co 7133  Topctop 21477  clsccl 21602  neicnei 21681  Hauscha 21892  Regcreg 21893   ×t ctx 22144  UnifOncust 22784  unifTopcutop 22815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-iin 4898  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-er 8267  df-map 8386  df-en 8488  df-fin 8491  df-fi 8853  df-topgen 16696  df-top 21478  df-topon 21495  df-bases 21530  df-cld 21603  df-ntr 21604  df-cls 21605  df-nei 21682  df-cn 21811  df-cnp 21812  df-reg 21900  df-tx 22146  df-ust 22785  df-utop 22816
This theorem is referenced by:  uspreg  22859
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