ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  istps GIF version

Theorem istps 13720
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 13719 . . 3 TopSp = {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))}
21eleq2i 2244 . 2 (𝐾 ∈ TopSp ↔ 𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))})
3 topontop 13702 . . . 4 (𝐽 ∈ (TopOn‘𝐴) → 𝐽 ∈ Top)
4 topnfn 12699 . . . . . . 7 TopOpen Fn V
5 fnrel 5316 . . . . . . 7 (TopOpen Fn V → Rel TopOpen)
64, 5ax-mp 5 . . . . . 6 Rel TopOpen
7 0opn 13694 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ 𝐽)
8 istps.j . . . . . . 7 𝐽 = (TopOpen‘𝐾)
97, 8eleqtrdi 2270 . . . . . 6 (𝐽 ∈ Top → ∅ ∈ (TopOpen‘𝐾))
10 relelfvdm 5549 . . . . . 6 ((Rel TopOpen ∧ ∅ ∈ (TopOpen‘𝐾)) → 𝐾 ∈ dom TopOpen)
116, 9, 10sylancr 414 . . . . 5 (𝐽 ∈ Top → 𝐾 ∈ dom TopOpen)
1211elexd 2752 . . . 4 (𝐽 ∈ Top → 𝐾 ∈ V)
133, 12syl 14 . . 3 (𝐽 ∈ (TopOn‘𝐴) → 𝐾 ∈ V)
14 fveq2 5517 . . . . 5 (𝑓 = 𝐾 → (TopOpen‘𝑓) = (TopOpen‘𝐾))
1514, 8eqtr4di 2228 . . . 4 (𝑓 = 𝐾 → (TopOpen‘𝑓) = 𝐽)
16 fveq2 5517 . . . . . 6 (𝑓 = 𝐾 → (Base‘𝑓) = (Base‘𝐾))
17 istps.a . . . . . 6 𝐴 = (Base‘𝐾)
1816, 17eqtr4di 2228 . . . . 5 (𝑓 = 𝐾 → (Base‘𝑓) = 𝐴)
1918fveq2d 5521 . . . 4 (𝑓 = 𝐾 → (TopOn‘(Base‘𝑓)) = (TopOn‘𝐴))
2015, 19eleq12d 2248 . . 3 (𝑓 = 𝐾 → ((TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓)) ↔ 𝐽 ∈ (TopOn‘𝐴)))
2113, 20elab3 2891 . 2 (𝐾 ∈ {𝑓 ∣ (TopOpen‘𝑓) ∈ (TopOn‘(Base‘𝑓))} ↔ 𝐽 ∈ (TopOn‘𝐴))
222, 21bitri 184 1 (𝐾 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐴))
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1353  wcel 2148  {cab 2163  Vcvv 2739  c0 3424  dom cdm 4628  Rel wrel 4633   Fn wfn 5213  cfv 5218  Basecbs 12465  TopOpenctopn 12695  Topctop 13685  TopOnctopon 13698  TopSpctps 13718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-5 8984  df-6 8985  df-7 8986  df-8 8987  df-9 8988  df-ndx 12468  df-slot 12469  df-base 12471  df-tset 12558  df-rest 12696  df-topn 12697  df-top 13686  df-topon 13699  df-topsp 13719
This theorem is referenced by:  istps2  13721  tpspropd  13724  tsettps  13726  isxms2  14140
  Copyright terms: Public domain W3C validator