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Theorem sshauslem 21955
Description: Lemma for sshaus 21958 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 1133 . 2 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽𝐴)
2 f1oi 6625 . . 3 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1of1 6587 . . 3 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋1-1𝑋)
42, 3mp1i 13 . 2 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → ( I ↾ 𝑋):𝑋1-1𝑋)
5 simp3 1135 . . 3 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽𝐾)
6 simp2 1134 . . . 4 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾 ∈ (TopOn‘𝑋))
7 sshauslem.2 . . . . . 6 (𝐽𝐴𝐽 ∈ Top)
873ad2ant1 1130 . . . . 5 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽 ∈ Top)
9 t1sep.1 . . . . . 6 𝑋 = 𝐽
109toptopon 21500 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
118, 10sylib 221 . . . 4 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐽 ∈ (TopOn‘𝑋))
12 ssidcn 21838 . . . 4 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽) ↔ 𝐽𝐾))
136, 11, 12syl2anc 587 . . 3 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽) ↔ 𝐽𝐾))
145, 13mpbird 260 . 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 1368 1 ((𝐽𝐴𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2115  wss 3910   cuni 4811   I cid 5432  cres 5530  1-1wf1 6325  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7130  Topctop 21476  TopOnctopon 21493   Cn ccn 21807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-map 8383  df-top 21477  df-topon 21494  df-cn 21810
This theorem is referenced by:  sst0  21956  sst1  21957  sshaus  21958
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