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Theorem tususp 21886
 Description: A constructed uniform space is an uniform space. (Contributed by Thierry Arnoux, 5-Dec-2017.)
Hypothesis
Ref Expression
tuslem.k 𝐾 = (toUnifSp‘𝑈)
Assertion
Ref Expression
tususp (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 ∈ UnifSp)

Proof of Theorem tususp
StepHypRef Expression
1 id 22 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ∈ (UnifOn‘𝑋))
2 tuslem.k . . . 4 𝐾 = (toUnifSp‘𝑈)
32tususs 21884 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSt‘𝐾))
42tusbas 21882 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
54fveq2d 6107 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (UnifOn‘𝑋) = (UnifOn‘(Base‘𝐾)))
61, 3, 53eltr3d 2702 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (UnifSt‘𝐾) ∈ (UnifOn‘(Base‘𝐾)))
72tusunif 21883 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
87fveq2d 6107 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (unifTop‘(UnifSet‘𝐾)))
92tuslem 21881 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))
109simp3d 1068 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopOpen‘𝐾))
117, 3eqtr3d 2646 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSt‘𝐾))
1211fveq2d 6107 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘(UnifSet‘𝐾)) = (unifTop‘(UnifSt‘𝐾)))
138, 10, 123eqtr3d 2652 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (TopOpen‘𝐾) = (unifTop‘(UnifSt‘𝐾)))
14 eqid 2610 . . 3 (Base‘𝐾) = (Base‘𝐾)
15 eqid 2610 . . 3 (UnifSt‘𝐾) = (UnifSt‘𝐾)
16 eqid 2610 . . 3 (TopOpen‘𝐾) = (TopOpen‘𝐾)
1714, 15, 16isusp 21875 . 2 (𝐾 ∈ UnifSp ↔ ((UnifSt‘𝐾) ∈ (UnifOn‘(Base‘𝐾)) ∧ (TopOpen‘𝐾) = (unifTop‘(UnifSt‘𝐾))))
186, 13, 17sylanbrc 695 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 ∈ UnifSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  Basecbs 15695  UnifSetcunif 15778  TopOpenctopn 15905  UnifOncust 21813  unifTopcutop 21844  UnifStcuss 21867  UnifSpcusp 21868  toUnifSpctus 21869 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-tset 15787  df-unif 15792  df-rest 15906  df-topn 15907  df-ust 21814  df-utop 21845  df-uss 21870  df-usp 21871  df-tus 21872 This theorem is referenced by:  cmetcusp  22958
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