Theorem hmphtr 23773

Description: "Is homeomorphic to" is transitive. (Contributed by FL, 9-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
hmphtr	⊢ ((𝐽 ≃ 𝐾 ∧ 𝐾 ≃ 𝐿) → 𝐽 ≃ 𝐿)

Proof of Theorem hmphtr

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmph 23766	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		hmph 23766	. 2 ⊢ (𝐾 ≃ 𝐿 ↔ (𝐾Homeo𝐿) ≠ ∅)
3		n0 4288	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
4		n0 4288	. . 3 ⊢ ((𝐾Homeo𝐿) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿))
5		exdistrv 1962	. . . 4 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) ↔ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)))
6		hmeoco 23762	. . . . . 6 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → (𝑔 ∘ 𝑓) ∈ (𝐽Homeo𝐿))
7		hmphi 23767	. . . . . 6 ⊢ ((𝑔 ∘ 𝑓) ∈ (𝐽Homeo𝐿) → 𝐽 ≃ 𝐿)
8	6, 7	syl 17	. . . . 5 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽 ≃ 𝐿)
9	8	exlimivv 1939	. . . 4 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽 ≃ 𝐿)
10	5, 9	sylbir 236	. . 3 ⊢ ((∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽 ≃ 𝐿)
11	3, 4, 10	syl2anb 604	. 2 ⊢ (((𝐽Homeo𝐾) ≠ ∅ ∧ (𝐾Homeo𝐿) ≠ ∅) → 𝐽 ≃ 𝐿)
12	1, 2, 11	syl2anb 604	1 ⊢ ((𝐽 ≃ 𝐾 ∧ 𝐾 ≃ 𝐿) → 𝐽 ≃ 𝐿)

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1786   ∈ wcel 2119   ≠ wne 2935  ∅c0 4268   class class class wbr 5079   ∘ ccom 5629  (class class class)co 7363  Homeochmeo 23743   ≃ chmph 23744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-1o 8402  df-map 8772  df-top 22884  df-topon 22901  df-cn 23217  df-hmeo 23745  df-hmph 23746

This theorem is referenced by:  hmpher  23774  xrhmph  24939