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Theorem haushmphlem 23136
Description: Lemma for haushmph 23141 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 23131 . 2 (𝐽𝐾𝐾𝐽)
2 hmph 23125 . . 3 (𝐾𝐽 ↔ (𝐾Homeo𝐽) ≠ ∅)
3 n0 4306 . . . 4 ((𝐾Homeo𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐾Homeo𝐽))
4 simpl 483 . . . . . . 7 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝐽𝐴)
5 eqid 2736 . . . . . . . . . 10 𝐾 = 𝐾
6 eqid 2736 . . . . . . . . . 10 𝐽 = 𝐽
75, 6hmeof1o 23113 . . . . . . . . 9 (𝑓 ∈ (𝐾Homeo𝐽) → 𝑓: 𝐾1-1-onto 𝐽)
87adantl 482 . . . . . . . 8 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓: 𝐾1-1-onto 𝐽)
9 f1of1 6783 . . . . . . . 8 (𝑓: 𝐾1-1-onto 𝐽𝑓: 𝐾1-1 𝐽)
108, 9syl 17 . . . . . . 7 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓: 𝐾1-1 𝐽)
11 hmeocn 23109 . . . . . . . 8 (𝑓 ∈ (𝐾Homeo𝐽) → 𝑓 ∈ (𝐾 Cn 𝐽))
1211adantl 482 . . . . . . 7 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓 ∈ (𝐾 Cn 𝐽))
13 haushmphlem.2 . . . . . . 7 ((𝐽𝐴𝑓: 𝐾1-1 𝐽𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾𝐴)
144, 10, 12, 13syl3anc 1371 . . . . . 6 ((𝐽𝐴𝑓 ∈ (𝐾Homeo𝐽)) → 𝐾𝐴)
1514expcom 414 . . . . 5 (𝑓 ∈ (𝐾Homeo𝐽) → (𝐽𝐴𝐾𝐴))
1615exlimiv 1933 . . . 4 (∃𝑓 𝑓 ∈ (𝐾Homeo𝐽) → (𝐽𝐴𝐾𝐴))
173, 16sylbi 216 . . 3 ((𝐾Homeo𝐽) ≠ ∅ → (𝐽𝐴𝐾𝐴))
182, 17sylbi 216 . 2 (𝐾𝐽 → (𝐽𝐴𝐾𝐴))
191, 18syl 17 1 (𝐽𝐾 → (𝐽𝐴𝐾𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087  wex 1781  wcel 2106  wne 2943  c0 4282   cuni 4865   class class class wbr 5105  1-1wf1 6493  1-1-ontowf1o 6495  (class class class)co 7356  Topctop 22240   Cn ccn 22573  Homeochmeo 23102  chmph 23103
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7920  df-2nd 7921  df-1o 8411  df-map 8766  df-top 22241  df-topon 22258  df-cn 22576  df-hmeo 23104  df-hmph 23105
This theorem is referenced by:  t0hmph  23139  t1hmph  23140  haushmph  23141
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