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Theorem haushmphlem 21400
 Description: Lemma for haushmph 21405 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 21395 . 2 (𝐽𝐾𝐾𝐽)
2 hmph 21389 . . 3 (𝐾𝐽 ↔ (𝐾Homeo𝐽) ≠ ∅)
3 n0 3890 . . . 4 ((𝐾Homeo𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐾Homeo𝐽))
4 simpl 472 . . . . . . 7 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝐽𝐴)
5 eqid 2610 . . . . . . . . . 10 𝐾 = 𝐾
6 eqid 2610 . . . . . . . . . 10 𝐽 = 𝐽
75, 6hmeof1o 21377 . . . . . . . . 9 (𝑓 ∈ (𝐾Homeo𝐽) → 𝑓: 𝐾1-1-onto 𝐽)
87adantl 481 . . . . . . . 8 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓: 𝐾1-1-onto 𝐽)
9 f1of1 6049 . . . . . . . 8 (𝑓: 𝐾1-1-onto 𝐽𝑓: 𝐾1-1 𝐽)
108, 9syl 17 . . . . . . 7 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓: 𝐾1-1 𝐽)
11 hmeocn 21373 . . . . . . . 8 (𝑓 ∈ (𝐾Homeo𝐽) → 𝑓 ∈ (𝐾 Cn 𝐽))
1211adantl 481 . . . . . . 7 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓 ∈ (𝐾 Cn 𝐽))
13 haushmphlem.2 . . . . . . 7 ((𝐽𝐴𝑓: 𝐾1-1 𝐽𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾𝐴)
144, 10, 12, 13syl3anc 1318 . . . . . 6 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝐾𝐴)
1514expcom 450 . . . . 5 (𝑓 ∈ (𝐾Homeo𝐽) → (𝐽𝐴𝐾𝐴))
1615exlimiv 1845 . . . 4 (∃𝑓 𝑓 ∈ (𝐾Homeo𝐽) → (𝐽𝐴𝐾𝐴))
173, 16sylbi 206 . . 3 ((𝐾Homeo𝐽) ≠ ∅ → (𝐽𝐴𝐾𝐴))
182, 17sylbi 206 . 2 (𝐾𝐽 → (𝐽𝐴𝐾𝐴))
191, 18syl 17 1 (𝐽𝐾 → (𝐽𝐴𝐾𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∅c0 3874  ∪ cuni 4372   class class class wbr 4583  –1-1→wf1 5801  –1-1-onto→wf1o 5803  (class class class)co 6549  Topctop 20517   Cn ccn 20838  Homeochmeo 21366   ≃ chmph 21367 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-1o 7447  df-map 7746  df-top 20521  df-topon 20523  df-cn 20841  df-hmeo 21368  df-hmph 21369 This theorem is referenced by:  t0hmph  21403  t1hmph  21404  haushmph  21405
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