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Theorem hmphtr 23211
Description: "Is homeomorphic to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
hmphtr ((𝐽𝐾𝐾𝐿) → 𝐽𝐿)

Proof of Theorem hmphtr
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hmph 23204 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 hmph 23204 . 2 (𝐾𝐿 ↔ (𝐾Homeo𝐿) ≠ ∅)
3 n0 4339 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
4 n0 4339 . . 3 ((𝐾Homeo𝐿) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿))
5 exdistrv 1959 . . . 4 (∃𝑓𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) ↔ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)))
6 hmeoco 23200 . . . . . 6 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → (𝑔𝑓) ∈ (𝐽Homeo𝐿))
7 hmphi 23205 . . . . . 6 ((𝑔𝑓) ∈ (𝐽Homeo𝐿) → 𝐽𝐿)
86, 7syl 17 . . . . 5 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽𝐿)
98exlimivv 1935 . . . 4 (∃𝑓𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽𝐿)
105, 9sylbir 234 . . 3 ((∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽𝐿)
113, 4, 10syl2anb 598 . 2 (((𝐽Homeo𝐾) ≠ ∅ ∧ (𝐾Homeo𝐿) ≠ ∅) → 𝐽𝐿)
121, 2, 11syl2anb 598 1 ((𝐽𝐾𝐾𝐿) → 𝐽𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1781  wcel 2106  wne 2939  c0 4315   class class class wbr 5138  ccom 5670  (class class class)co 7390  Homeochmeo 23181  chmph 23182
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 2702  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7705
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3430  df-v 3472  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-suc 6356  df-iota 6481  df-fun 6531  df-fn 6532  df-f 6533  df-fv 6537  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7954  df-2nd 7955  df-1o 8445  df-map 8802  df-top 22320  df-topon 22337  df-cn 22655  df-hmeo 23183  df-hmph 23184
This theorem is referenced by:  hmpher  23212  xrhmph  24387
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