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Theorem hmphtr 21786
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 21779 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 hmph 21779 . 2 (𝐾𝐿 ↔ (𝐾Homeo𝐿) ≠ ∅)
3 n0 4072 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
4 n0 4072 . . 3 ((𝐾Homeo𝐿) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿))
5 eeanv 2325 . . . 4 (∃𝑓𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) ↔ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)))
6 hmeoco 21775 . . . . . 6 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → (𝑔𝑓) ∈ (𝐽Homeo𝐿))
7 hmphi 21780 . . . . . 6 ((𝑔𝑓) ∈ (𝐽Homeo𝐿) → 𝐽𝐿)
86, 7syl 17 . . . . 5 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽𝐿)
98exlimivv 2007 . . . 4 (∃𝑓𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽𝐿)
105, 9sylbir 225 . . 3 ((∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽𝐿)
113, 4, 10syl2anb 497 . 2 (((𝐽Homeo𝐾) ≠ ∅ ∧ (𝐾Homeo𝐿) ≠ ∅) → 𝐽𝐿)
121, 2, 11syl2anb 497 1 ((𝐽𝐾𝐾𝐿) → 𝐽𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wex 1851  wcel 2137  wne 2930  c0 4056   class class class wbr 4802  ccom 5268  (class class class)co 6811  Homeochmeo 21756  chmph 21757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-1st 7331  df-2nd 7332  df-1o 7727  df-map 8023  df-top 20899  df-topon 20916  df-cn 21231  df-hmeo 21758  df-hmph 21759
This theorem is referenced by:  hmpher  21787  xrhmph  22945
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