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Theorem istps 12239
 Description: Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
istps.a 𝐴 = (Base‘𝐾)
istps.j 𝐽 = (TopOpen‘𝐾)
Assertion
Ref Expression
istps (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))

Proof of Theorem istps
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 df-topsp 12238 . . 3 TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
21eleq2i 2207 . 2 (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))})
3 topontop 12221 . . . 4 (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top)
4 topnfn 12165 . . . . . . 7 TopOpen Fn V
5 fnrel 5229 . . . . . . 7 (TopOpen Fn V → Rel TopOpen)
64, 5ax-mp 5 . . . . . 6 Rel TopOpen
7 0opn 12213 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ 𝐽)
8 istps.j . . . . . . 7 𝐽 = (TopOpen‘𝐾)
97, 8eleqtrdi 2233 . . . . . 6 (𝐽 ∈ Top → ∅ ∈ (TopOpen‘𝐾))
10 relelfvdm 5461 . . . . . 6 ((Rel TopOpen ∧ ∅ ∈ (TopOpen‘𝐾)) → 𝐾 ∈ dom TopOpen)
116, 9, 10sylancr 411 . . . . 5 (𝐽 ∈ Top → 𝐾 ∈ dom TopOpen)
1211elexd 2702 . . . 4 (𝐽 ∈ Top → 𝐾 ∈ V)
133, 12syl 14 . . 3 (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ V)
14 fveq2 5429 . . . . 5 (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
1514, 8eqtr4di 2191 . . . 4 (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
16 fveq2 5429 . . . . . 6 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
17 istps.a . . . . . 6 𝐴 = (Base‘𝐾)
1816, 17eqtr4di 2191 . . . . 5 (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴)
1918fveq2d 5433 . . . 4 (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴))
2015, 19eleq12d 2211 . . 3 (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴)))
2113, 20elab3 2840 . 2 (𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} ↔ 𝐽 ∈ (TopOn‘𝐴))
222, 21bitri 183 1 (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1332   ∈ wcel 1481  {cab 2126  Vcvv 2689  ∅c0 3368  dom cdm 4547  Rel wrel 4552   Fn wfn 5126  ‘cfv 5131  Basecbs 11999  TopOpenctopn 12161  Topctop 12204  TopOnctopon 12217  TopSpctps 12237 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-cnex 7736  ax-resscn 7737  ax-1re 7739  ax-addrcl 7742 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-inn 8746  df-2 8804  df-3 8805  df-4 8806  df-5 8807  df-6 8808  df-7 8809  df-8 8810  df-9 8811  df-ndx 12002  df-slot 12003  df-base 12005  df-tset 12080  df-rest 12162  df-topn 12163  df-top 12205  df-topon 12218  df-topsp 12238 This theorem is referenced by:  istps2  12240  tpspropd  12243  tsettps  12245  isxms2  12661
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