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Theorem tpspropd 12130
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 5393 . . 3 (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿)))
41, 3eleq12d 2188 . 2 (𝜑 → ((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿))))
5 eqid 2117 . . 3 (Base‘𝐾) = (Base‘𝐾)
6 eqid 2117 . . 3 (TopOpen‘𝐾) = (TopOpen‘𝐾)
75, 6istps 12126 . 2 (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
8 eqid 2117 . . 3 (Base‘𝐿) = (Base‘𝐿)
9 eqid 2117 . . 3 (TopOpen‘𝐿) = (TopOpen‘𝐿)
108, 9istps 12126 . 2 (𝐿 ∈ TopSp ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿)))
114, 7, 103bitr4g 222 1 (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  wcel 1465  cfv 5093  Basecbs 11886  TopOpenctopn 12048  TopOnctopon 12104  TopSpctps 12124
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-inn 8689  df-2 8747  df-3 8748  df-4 8749  df-5 8750  df-6 8751  df-7 8752  df-8 8753  df-9 8754  df-ndx 11889  df-slot 11890  df-base 11892  df-tset 11967  df-rest 12049  df-topn 12050  df-top 12092  df-topon 12105  df-topsp 12125
This theorem is referenced by:  xmspropd  12573
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