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Theorem utopreg 22433
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 22415 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
32adantr 474 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → (unifTop‘𝑈) ∈ Top)
41, 3syl5eqel 2910 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Top)
5 simp-4l 801 . . . . . . . . 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 717 . . . . . . . . 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 785 . . . . . . . . 9 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑤𝑈)
9 simp-4l 801 . . . . . . . . . 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 721 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝐽 ∈ Top)
12 simpllr 793 . . . . . . . . . . . . 13 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎𝐽)
13 eqid 2825 . . . . . . . . . . . . . 14 𝐽 = 𝐽
1413eltopss 21089 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑎𝐽) → 𝑎 𝐽)
1511, 12, 14syl2anc 579 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎 𝐽)
16 utopbas 22416 . . . . . . . . . . . . . 14 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
171unieqi 4669 . . . . . . . . . . . . . 14 𝐽 = (unifTop‘𝑈)
1816, 17syl6eqr 2879 . . . . . . . . . . . . 13 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = 𝐽)
199, 18syl 17 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑋 = 𝐽)
2015, 19sseqtr4d 3867 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑎𝑋)
21 simplr 785 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑥𝑎)
2220, 21sseldd 3828 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → 𝑥𝑋)
231utopsnnei 22430 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑥𝑋) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
249, 10, 22, 23syl3anc 1494 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑤𝑈) → (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥}))
255, 8, 24syl2anc 579 . . . . . . . 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 21290 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑏𝐽 ({𝑥} ⊆ 𝑏𝑏 ⊆ (𝑤 “ {𝑥})))
277, 25, 26syl2anc 579 . . . . . . 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 787 . . . . . . . . . . 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 3417 . . . . . . . . . . . 12 𝑥 ∈ V
3029snss 4537 . . . . . . . . . . 11 (𝑥𝑏 ↔ {𝑥} ⊆ 𝑏)
3128, 30sylibr 226 . . . . . . . . . 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 717 . . . . . . . . . . . . 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 791 . . . . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . 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 785 . . . . . . . . . . . . . . . . . . 19 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎𝐽)
386, 37, 14syl2anc 579 . . . . . . . . . . . . . . . . . 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 3867 . . . . . . . . . . . . . . . . 17 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎𝑋)
41 simpr 479 . . . . . . . . . . . . . . . . 17 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑥𝑎)
4240, 41sseldd 3828 . . . . . . . . . . . . . . . 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 22409 . . . . . . . . . . . . . . 15 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈𝑥𝑋) → (𝑤 “ {𝑥}) ⊆ 𝑋)
4535, 36, 43, 44syl3anc 1494 . . . . . . . . . . . . . 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 3866 . . . . . . . . . . . . 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 789 . . . . . . . . . . . . 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 21236 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ (𝑤 “ {𝑥}) ⊆ 𝐽𝑏 ⊆ (𝑤 “ {𝑥})) → ((cls‘𝐽)‘𝑏) ⊆ ((cls‘𝐽)‘(𝑤 “ {𝑥})))
5032, 47, 48, 49syl3anc 1494 . . . . . . . . . . . 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 22385 . . . . . . . . . . . . . . . . 17 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤𝑈) → 𝑤 ⊆ (𝑋 × 𝑋))
5234, 8, 51syl2anc 579 . . . . . . . . . . . . . . . 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 5378 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (𝑋 × 𝑋) = ( 𝐽 × 𝐽))
5552, 54sseqtrd 3866 . . . . . . . . . . . . . . 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 807 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑥𝑎)
5856, 57sseldd 3828 . . . . . . . . . . . . . . 15 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑥 𝐽)
5913, 13imasncls 21873 . . . . . . . . . . . . . . 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 872 . . . . . . . . . . . . . 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 787 . . . . . . . . . . . . . . . . 17 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → 𝑤 = 𝑤)
621utop3cls 22432 . . . . . . . . . . . . . . . . 17 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ⊆ (𝑋 × 𝑋)) ∧ (𝑤𝑈𝑤 = 𝑤)) → ((cls‘(𝐽 ×t 𝐽))‘𝑤) ⊆ (𝑤 ∘ (𝑤𝑤)))
6334, 52, 8, 61, 62syl22anc 872 . . . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . . 16 ((((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) ∧ 𝑤𝑈) ∧ (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)) → (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣)
6563, 64sstrd 3837 . . . . . . . . . . . . . . 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 5745 . . . . . . . . . . . . . . 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 3837 . . . . . . . . . . . . 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 717 . . . . . . . . . . . 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 3837 . . . . . . . . . . 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 807 . . . . . . . . . . 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 3867 . . . . . . . . . 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 507 . . . . . . . . 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 403 . . . . . . . 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 3225 . . . . . . 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 805 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑈 ∈ (UnifOn‘𝑋))
78 simplr 785 . . . . . . 7 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → 𝑣𝑈)
79 ustex3sym 22398 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑣𝑈) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣))
8077, 78, 79syl2anc 579 . . . . . 6 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → ∃𝑤𝑈 (𝑤 = 𝑤 ∧ (𝑤 ∘ (𝑤𝑤)) ⊆ 𝑣))
8176, 80r19.29a 3288 . . . . 5 ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) ∧ 𝑣𝑈) ∧ 𝑎 = (𝑣 “ {𝑥})) → ∃𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
82 opnneip 21301 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑎𝐽𝑥𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
836, 37, 41, 82syl3anc 1494 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎 ∈ ((nei‘𝐽)‘{𝑥}))
841utopsnneip 22429 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥𝑋) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})))
8533, 42, 84syl2anc 579 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ((nei‘𝐽)‘{𝑥}) = ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})))
8683, 85eleqtrd 2908 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → 𝑎 ∈ ran (𝑣𝑈 ↦ (𝑣 “ {𝑥})))
87 eqid 2825 . . . . . . . 8 (𝑣𝑈 ↦ (𝑣 “ {𝑥})) = (𝑣𝑈 ↦ (𝑣 “ {𝑥}))
8887elrnmpt 5609 . . . . . . 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 224 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ∃𝑣𝑈 𝑎 = (𝑣 “ {𝑥}))
9181, 90r19.29a 3288 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) ∧ 𝑥𝑎) → ∃𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
9291ralrimiva 3175 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) ∧ 𝑎𝐽) → ∀𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
9392ralrimiva 3175 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → ∀𝑎𝐽𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎))
94 isreg 21514 . 2 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑎𝐽𝑥𝑎𝑏𝐽 (𝑥𝑏 ∧ ((cls‘𝐽)‘𝑏) ⊆ 𝑎)))
954, 93, 94sylanbrc 578 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝐽 ∈ Haus) → 𝐽 ∈ Reg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  wral 3117  wrex 3118  wss 3798  {csn 4399   cuni 4660  cmpt 4954   × cxp 5344  ccnv 5345  ran crn 5347  cima 5349  ccom 5350  cfv 6127  (class class class)co 6910  Topctop 21075  clsccl 21200  neicnei 21279  Hauscha 21490  Regcreg 21491   ×t ctx 21741  UnifOncust 22380  unifTopcutop 22411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-er 8014  df-map 8129  df-en 8229  df-fin 8232  df-fi 8592  df-topgen 16464  df-top 21076  df-topon 21093  df-bases 21128  df-cld 21201  df-ntr 21202  df-cls 21203  df-nei 21280  df-cn 21409  df-cnp 21410  df-reg 21498  df-tx 21743  df-ust 22381  df-utop 22412
This theorem is referenced by:  uspreg  22455
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