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Theorem tpspropd 13194
Description: A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
tpspropd.1 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
tpspropd.2 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
Assertion
Ref Expression
tpspropd (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))

Proof of Theorem tpspropd
StepHypRef Expression
1 tpspropd.2 . . 3 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
2 tpspropd.1 . . . 4 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
32fveq2d 5515 . . 3 (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿)))
41, 3eleq12d 2248 . 2 (𝜑 → ((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿))))
5 eqid 2177 . . 3 (Base‘𝐾) = (Base‘𝐾)
6 eqid 2177 . . 3 (TopOpen‘𝐾) = (TopOpen‘𝐾)
75, 6istps 13190 . 2 (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
8 eqid 2177 . . 3 (Base‘𝐿) = (Base‘𝐿)
9 eqid 2177 . . 3 (TopOpen‘𝐿) = (TopOpen‘𝐿)
108, 9istps 13190 . 2 (𝐿 ∈ TopSp ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿)))
114, 7, 103bitr4g 223 1 (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  cfv 5212  Basecbs 12442  TopOpenctopn 12634  TopOnctopon 13168  TopSpctps 13188
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 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-cnex 7890  ax-resscn 7891  ax-1re 7893  ax-addrcl 7896
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-inn 8906  df-2 8964  df-3 8965  df-4 8966  df-5 8967  df-6 8968  df-7 8969  df-8 8970  df-9 8971  df-ndx 12445  df-slot 12446  df-base 12448  df-tset 12534  df-rest 12635  df-topn 12636  df-top 13156  df-topon 13169  df-topsp 13189
This theorem is referenced by:  xmspropd  13637
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