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Theorem utopreg 22277
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 22259 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
32adantr 472 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → (unifTop‘𝑈) ∈ Top)
41, 3syl5eqel 2843 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
5 simp-4l 825 . . . . . . . . 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 764 . . . . . . . . 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 809 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑤𝑈)
9 simp-4l 825 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑈 ∈ (UnifOn‘𝑋))
10 simpr 479 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑤𝑈)
114ad3antrrr 768 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝐽 ∈ Top)
12 simpllr 817 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎𝐽)
13 eqid 2760 . . . . . . . . . . . . . 14 𝐽 = 𝐽
1413eltopss 20934 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑎𝐽) → 𝑎 𝐽)
1511, 12, 14syl2anc 696 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎 𝐽)
16 utopbas 22260 . . . . . . . . . . . . . 14 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
171unieqi 4597 . . . . . . . . . . . . . 14 𝐽 = (unifTop‘𝑈)
1816, 17syl6eqr 2812 . . . . . . . . . . . . 13 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = 𝐽)
199, 18syl 17 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑋 = 𝐽)
2015, 19sseqtr4d 3783 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎𝑋)
21 simplr 809 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑥𝑎)
2220, 21sseldd 3745 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑥𝑋)
231utopsnnei 22274 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑥𝑋) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
249, 10, 22, 23syl3anc 1477 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
255, 8, 24syl2anc 696 . . . . . . . 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 21134 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑏𝐽 ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥})))
277, 25, 26syl2anc 696 . . . . . . 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 811 . . . . . . . . . . 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 3343 . . . . . . . . . . . 12 𝑥 ∈ V
3029snss 4460 . . . . . . . . . . 11 (𝑥𝑏 ↔ {𝑥} ⊆ 𝑏)
3128, 30sylibr 224 . . . . . . . . . 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 764 . . . . . . . . . . . . 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 815 . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . 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 809 . . . . . . . . . . . . . . . . . . 19 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎𝐽)
386, 37, 14syl2anc 696 . . . . . . . . . . . . . . . . . 18 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎 𝐽)
3933, 18syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑋 = 𝐽)
4038, 39sseqtr4d 3783 . . . . . . . . . . . . . . . . 17 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎𝑋)
41 simpr 479 . . . . . . . . . . . . . . . . 17 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑥𝑎)
4240, 41sseldd 3745 . . . . . . . . . . . . . . . 16 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑥𝑋)
4342ad6antr 779 . . . . . . . . . . . . . . 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 22253 . . . . . . . . . . . . . . 15 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑥𝑋) → (𝑤 “ {𝑥}) ⊆ 𝑋)
4535, 36, 43, 44syl3anc 1477 . . . . . . . . . . . . . 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 3782 . . . . . . . . . . . . 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 813 . . . . . . . . . . . . 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 21080 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ⊆ 𝐽𝑏 ⊆ (𝑤 “ {𝑥})) → ((cls‘𝐽)‘𝑏) ⊆ ((cls‘𝐽)‘(𝑤 “ {𝑥})))
5032, 47, 48, 49syl3anc 1477 . . . . . . . . . . . 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 22229 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
5234, 8, 51syl2anc 696 . . . . . . . . . . . . . . . 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 5298 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (𝑋 × 𝑋) = ( 𝐽 × 𝐽))
5552, 54sseqtrd 3782 . . . . . . . . . . . . . . 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 831 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑥𝑎)
5856, 57sseldd 3745 . . . . . . . . . . . . . . 15 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑥 𝐽)
5913, 13imasncls 21717 . . . . . . . . . . . . . . 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 1478 . . . . . . . . . . . . . 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 811 . . . . . . . . . . . . . . . . 17 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑤 = 𝑤)
621utop3cls 22276 . . . . . . . . . . . . . . . . 17 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ⊆ (𝑋 × 𝑋)) ∧ (𝑤𝑈𝑤 = 𝑤)) → ((cls‘(𝐽 ×t 𝐽))‘𝑤) ⊆ (𝑤 ∘ (𝑤𝑤)))
6334, 52, 8, 61, 62syl22anc 1478 . . . . . . . . . . . . . . . 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 813 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)
6563, 64sstrd 3754 . . . . . . . . . . . . . . 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 5658 . . . . . . . . . . . . . . 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 3754 . . . . . . . . . . . . 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 764 . . . . . . . . . . . 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 3754 . . . . . . . . . . 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 831 . . . . . . . . . . 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, 71sseqtr4d 3783 . . . . . . . . . 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 555 . . . . . . . . 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 449 . . . . . . . 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 3155 . . . . . . 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 829 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑈 ∈ (UnifOn‘𝑋))
78 simplr 809 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑣𝑈)
79 ustex3sym 22242 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣))
8077, 78, 79syl2anc 696 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣))
8176, 80r19.29a 3216 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → ∃𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
82 opnneip 21145 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑎𝐽𝑥𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
836, 37, 41, 82syl3anc 1477 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
841utopsnneip 22273 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑋) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})))
8533, 42, 84syl2anc 696 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})))
8683, 85eleqtrd 2841 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})))
87 eqid 2760 . . . . . . . 8 (𝑣𝑈 ↦ (𝑣 “ {𝑥})) = (𝑣𝑈 ↦ (𝑣 “ {𝑥}))
8887elrnmpt 5527 . . . . . . 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 222 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑥}))
9181, 90r19.29a 3216 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ∃𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
9291ralrimiva 3104 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) → ∀𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
9392ralrimiva 3104 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → ∀𝑎𝐽𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
94 isreg 21358 . 2 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑎𝐽𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎)))
954, 93, 94sylanbrc 701 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  wss 3715  {csn 4321   cuni 4588  cmpt 4881   × cxp 5264  ccnv 5265  ran crn 5267  cima 5269  ccom 5270  cfv 6049  (class class class)co 6814  Topctop 20920  clsccl 21044  neicnei 21123  Hauscha 21334  Regcreg 21335   ×t ctx 21585  UnifOncust 22224  unifTopcutop 22255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-en 8124  df-fin 8127  df-fi 8484  df-topgen 16326  df-top 20921  df-topon 20938  df-bases 20972  df-cld 21045  df-ntr 21046  df-cls 21047  df-nei 21124  df-cn 21253  df-cnp 21254  df-reg 21342  df-tx 21587  df-ust 22225  df-utop 22256
This theorem is referenced by:  uspreg  22299
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