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Theorem sshauslem 21908
Description: Lemma for sshaus 21911 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 1128 . 2 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽𝐴)
2 f1oi 6645 . . 3 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1of1 6607 . . 3 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋1-1𝑋)
42, 3mp1i 13 . 2 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → ( I ↾ 𝑋):𝑋1-1𝑋)
5 simp3 1130 . . 3 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽𝐾)
6 simp2 1129 . . . 4 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾 ∈ (TopOn‘𝑋))
7 sshauslem.2 . . . . . 6 (𝐽𝐴𝐽 ∈ Top)
873ad2ant1 1125 . . . . 5 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽 ∈ Top)
9 t1sep.1 . . . . . 6 𝑋 = 𝐽
109toptopon 21453 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
118, 10sylib 219 . . . 4 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽 ∈ (TopOn‘𝑋))
12 ssidcn 21791 . . . 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 258 . 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 1363 1 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3a 1079   = wceq 1528  wcel 2105  wss 3933   cuni 4830   I cid 5452  cres 5550  1-1wf1 6345  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  Topctop 21429  TopOnctopon 21446   Cn ccn 21760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-top 21430  df-topon 21447  df-cn 21763
This theorem is referenced by:  sst0  21909  sst1  21910  sshaus  21911
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