MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sshauslem Structured version   Visualization version   GIF version

Theorem sshauslem 22533
Description: Lemma for sshaus 22536 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then a topology finer than one with property 𝐴 also has property 𝐴. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypotheses
Ref Expression
t1sep.1 𝑋 = 𝐽
sshauslem.2 (𝐽𝐴𝐽 ∈ Top)
sshauslem.3 ((𝐽𝐴 ∧ ( I ↾ 𝑋):𝑋1-1𝑋 ∧ ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽)) → 𝐾𝐴)
Assertion
Ref Expression
sshauslem ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾𝐴)

Proof of Theorem sshauslem
StepHypRef Expression
1 simp1 1135 . 2 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽𝐴)
2 f1oi 6746 . . 3 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1of1 6707 . . 3 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋1-1𝑋)
42, 3mp1i 13 . 2 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → ( I ↾ 𝑋):𝑋1-1𝑋)
5 simp3 1137 . . 3 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽𝐾)
6 simp2 1136 . . . 4 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾 ∈ (TopOn‘𝑋))
7 sshauslem.2 . . . . . 6 (𝐽𝐴𝐽 ∈ Top)
873ad2ant1 1132 . . . . 5 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽 ∈ Top)
9 t1sep.1 . . . . . 6 𝑋 = 𝐽
109toptopon 22076 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
118, 10sylib 217 . . . 4 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽 ∈ (TopOn‘𝑋))
12 ssidcn 22416 . . . 4 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽) ↔ 𝐽𝐾))
136, 11, 12syl2anc 584 . . 3 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽) ↔ 𝐽𝐾))
145, 13mpbird 256 . 2 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽))
15 sshauslem.3 . 2 ((𝐽𝐴 ∧ ( I ↾ 𝑋):𝑋1-1𝑋 ∧ ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽)) → 𝐾𝐴)
161, 4, 14, 15syl3anc 1370 1 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1086   = wceq 1539  wcel 2106  wss 3886   cuni 4839   I cid 5483  cres 5586  1-1wf1 6423  1-1-ontowf1o 6425  cfv 6426  (class class class)co 7267  Topctop 22052  TopOnctopon 22069   Cn ccn 22385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-sbc 3716  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-ov 7270  df-oprab 7271  df-mpo 7272  df-map 8604  df-top 22053  df-topon 22070  df-cn 22388
This theorem is referenced by:  sst0  22534  sst1  22535  sshaus  22536
  Copyright terms: Public domain W3C validator