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Theorem hmphtr 22386
 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 22379 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 hmph 22379 . 2 (𝐾𝐿 ↔ (𝐾Homeo𝐿) ≠ ∅)
3 n0 4282 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
4 n0 4282 . . 3 ((𝐾Homeo𝐿) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿))
5 exdistrv 1956 . . . 4 (∃𝑓𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) ↔ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)))
6 hmeoco 22375 . . . . . 6 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → (𝑔𝑓) ∈ (𝐽Homeo𝐿))
7 hmphi 22380 . . . . . 6 ((𝑔𝑓) ∈ (𝐽Homeo𝐿) → 𝐽𝐿)
86, 7syl 17 . . . . 5 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽𝐿)
98exlimivv 1933 . . . 4 (∃𝑓𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽𝐿)
105, 9sylbir 238 . . 3 ((∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽𝐿)
113, 4, 10syl2anb 600 . 2 (((𝐽Homeo𝐾) ≠ ∅ ∧ (𝐾Homeo𝐿) ≠ ∅) → 𝐽𝐿)
121, 2, 11syl2anb 600 1 ((𝐽𝐾𝐾𝐿) → 𝐽𝐿)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781   ∈ wcel 2114   ≠ wne 3011  ∅c0 4265   class class class wbr 5042   ∘ ccom 5536  (class class class)co 7140  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-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:  hmpher  22387  xrhmph  23550
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