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Theorem haushmphlem 22390
Description: Lemma for haushmph 22395 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then property 𝐴 is preserved under homeomorphisms. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypotheses
Ref Expression
haushmphlem.1 (𝐽𝐴𝐽 ∈ Top)
haushmphlem.2 ((𝐽𝐴𝑓: 𝐾1-1 𝐽𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾𝐴)
Assertion
Ref Expression
haushmphlem (𝐽𝐾 → (𝐽𝐴𝐾𝐴))
Distinct variable groups:   𝐴,𝑓   𝑓,𝐽   𝑓,𝐾

Proof of Theorem haushmphlem
StepHypRef Expression
1 hmphsym 22385 . 2 (𝐽𝐾𝐾𝐽)
2 hmph 22379 . . 3 (𝐾𝐽 ↔ (𝐾Homeo𝐽) ≠ ∅)
3 n0 4282 . . . 4 ((𝐾Homeo𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐾Homeo𝐽))
4 simpl 486 . . . . . . 7 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝐽𝐴)
5 eqid 2822 . . . . . . . . . 10 𝐾 = 𝐾
6 eqid 2822 . . . . . . . . . 10 𝐽 = 𝐽
75, 6hmeof1o 22367 . . . . . . . . 9 (𝑓 ∈ (𝐾Homeo𝐽) → 𝑓: 𝐾1-1-onto 𝐽)
87adantl 485 . . . . . . . 8 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓: 𝐾1-1-onto 𝐽)
9 f1of1 6596 . . . . . . . 8 (𝑓: 𝐾1-1-onto 𝐽𝑓: 𝐾1-1 𝐽)
108, 9syl 17 . . . . . . 7 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓: 𝐾1-1 𝐽)
11 hmeocn 22363 . . . . . . . 8 (𝑓 ∈ (𝐾Homeo𝐽) → 𝑓 ∈ (𝐾 Cn 𝐽))
1211adantl 485 . . . . . . 7 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓 ∈ (𝐾 Cn 𝐽))
13 haushmphlem.2 . . . . . . 7 ((𝐽𝐴𝑓: 𝐾1-1 𝐽𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾𝐴)
144, 10, 12, 13syl3anc 1368 . . . . . 6 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝐾𝐴)
1514expcom 417 . . . . 5 (𝑓 ∈ (𝐾Homeo𝐽) → (𝐽𝐴𝐾𝐴))
1615exlimiv 1931 . . . 4 (∃𝑓 𝑓 ∈ (𝐾Homeo𝐽) → (𝐽𝐴𝐾𝐴))
173, 16sylbi 220 . . 3 ((𝐾Homeo𝐽) ≠ ∅ → (𝐽𝐴𝐾𝐴))
182, 17sylbi 220 . 2 (𝐾𝐽 → (𝐽𝐴𝐾𝐴))
191, 18syl 17 1 (𝐽𝐾 → (𝐽𝐴𝐾𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wex 1781  wcel 2114  wne 3011  c0 4265   cuni 4813   class class class wbr 5042  1-1wf1 6331  1-1-ontowf1o 6333  (class class class)co 7140  Topctop 21496   Cn ccn 21827  Homeochmeo 22356  chmph 22357
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-1o 8089  df-map 8395  df-top 21497  df-topon 21514  df-cn 21830  df-hmeo 22358  df-hmph 22359
This theorem is referenced by:  t0hmph  22393  t1hmph  22394  haushmph  22395
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