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Theorem ismntop 33006
Description: Property of being a manifold. (Contributed by Thierry Arnoux, 5-Jan-2020.)
Assertion
Ref Expression
ismntop ((𝑁 ∈ β„•0 ∧ 𝐽 ∈ 𝑉) β†’ (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndΟ‰ ∧ 𝐽 ∈ Haus ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘))))))
Distinct variable groups:   𝑒,𝐽,π‘₯,𝑦   𝑒,𝑁,π‘₯,𝑦
Allowed substitution hints:   𝑉(π‘₯,𝑦,𝑒)

Proof of Theorem ismntop
StepHypRef Expression
1 ismntoplly 33005 . 2 ((𝑁 ∈ β„•0 ∧ 𝐽 ∈ 𝑉) β†’ (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndΟ‰ ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ )))
2 haustop 22835 . . . . . . . . 9 (𝐽 ∈ Haus β†’ 𝐽 ∈ Top)
32adantl 483 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ 𝐽 ∈ Haus) β†’ 𝐽 ∈ Top)
43biantrurd 534 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ 𝐽 ∈ Haus) β†’ (βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ ) ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ ))))
5 hmpher 23288 . . . . . . . . . . . . 13 ≃ Er Top
6 errel 8712 . . . . . . . . . . . . 13 ( ≃ Er Top β†’ Rel ≃ )
7 relelec 8748 . . . . . . . . . . . . 13 (Rel ≃ β†’ ((𝐽 β†Ύt 𝑒) ∈ [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ ↔ (TopOpenβ€˜(𝔼hilβ€˜π‘)) ≃ (𝐽 β†Ύt 𝑒)))
85, 6, 7mp2b 10 . . . . . . . . . . . 12 ((𝐽 β†Ύt 𝑒) ∈ [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ ↔ (TopOpenβ€˜(𝔼hilβ€˜π‘)) ≃ (𝐽 β†Ύt 𝑒))
9 hmphsymb 23290 . . . . . . . . . . . 12 ((TopOpenβ€˜(𝔼hilβ€˜π‘)) ≃ (𝐽 β†Ύt 𝑒) ↔ (𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘)))
108, 9bitr2i 276 . . . . . . . . . . 11 ((𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘)) ↔ (𝐽 β†Ύt 𝑒) ∈ [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ )
1110a1i 11 . . . . . . . . . 10 ((𝑁 ∈ β„•0 ∧ 𝐽 ∈ Haus) β†’ ((𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘)) ↔ (𝐽 β†Ύt 𝑒) ∈ [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ ))
1211anbi2d 630 . . . . . . . . 9 ((𝑁 ∈ β„•0 ∧ 𝐽 ∈ Haus) β†’ ((𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘))) ↔ (𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ )))
1312rexbidv 3179 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ 𝐽 ∈ Haus) β†’ (βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘))) ↔ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ )))
14132ralbidv 3219 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ 𝐽 ∈ Haus) β†’ (βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘))) ↔ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ )))
15 islly 22972 . . . . . . . 8 (𝐽 ∈ Locally [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ )))
1615a1i 11 . . . . . . 7 ((𝑁 ∈ β„•0 ∧ 𝐽 ∈ Haus) β†’ (𝐽 ∈ Locally [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ∈ [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ ))))
174, 14, 163bitr4rd 312 . . . . . 6 ((𝑁 ∈ β„•0 ∧ 𝐽 ∈ Haus) β†’ (𝐽 ∈ Locally [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ ↔ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘)))))
1817pm5.32da 580 . . . . 5 (𝑁 ∈ β„•0 β†’ ((𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ ) ↔ (𝐽 ∈ Haus ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘))))))
1918anbi2d 630 . . . 4 (𝑁 ∈ β„•0 β†’ ((𝐽 ∈ 2ndΟ‰ ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ )) ↔ (𝐽 ∈ 2ndΟ‰ ∧ (𝐽 ∈ Haus ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘)))))))
20 3anass 1096 . . . 4 ((𝐽 ∈ 2ndΟ‰ ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ ) ↔ (𝐽 ∈ 2ndΟ‰ ∧ (𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ )))
21 3anass 1096 . . . 4 ((𝐽 ∈ 2ndΟ‰ ∧ 𝐽 ∈ Haus ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘)))) ↔ (𝐽 ∈ 2ndΟ‰ ∧ (𝐽 ∈ Haus ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘))))))
2219, 20, 213bitr4g 314 . . 3 (𝑁 ∈ β„•0 β†’ ((𝐽 ∈ 2ndΟ‰ ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ ) ↔ (𝐽 ∈ 2ndΟ‰ ∧ 𝐽 ∈ Haus ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘))))))
2322adantr 482 . 2 ((𝑁 ∈ β„•0 ∧ 𝐽 ∈ 𝑉) β†’ ((𝐽 ∈ 2ndΟ‰ ∧ 𝐽 ∈ Haus ∧ 𝐽 ∈ Locally [(TopOpenβ€˜(𝔼hilβ€˜π‘))] ≃ ) ↔ (𝐽 ∈ 2ndΟ‰ ∧ 𝐽 ∈ Haus ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘))))))
241, 23bitrd 279 1 ((𝑁 ∈ β„•0 ∧ 𝐽 ∈ 𝑉) β†’ (𝑁ManTop𝐽 ↔ (𝐽 ∈ 2ndΟ‰ ∧ 𝐽 ∈ Haus ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (𝐽 ∩ 𝒫 π‘₯)(𝑦 ∈ 𝑒 ∧ (𝐽 β†Ύt 𝑒) ≃ (TopOpenβ€˜(𝔼hilβ€˜π‘))))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   ∩ cin 3948  π’« cpw 4603   class class class wbr 5149  Rel wrel 5682  β€˜cfv 6544  (class class class)co 7409   Er wer 8700  [cec 8701  β„•0cn0 12472   β†Ύt crest 17366  TopOpenctopn 17367  Topctop 22395  Hauscha 22812  2ndΟ‰c2ndc 22942  Locally clly 22968   ≃ chmph 23258  π”Όhilcehl 24901  ManTopcmntop 33002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-1o 8466  df-er 8703  df-ec 8705  df-map 8822  df-top 22396  df-topon 22413  df-cn 22731  df-haus 22819  df-lly 22970  df-hmeo 23259  df-hmph 23260  df-mntop 33003
This theorem is referenced by: (None)
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