Theorem hmphtr 23812

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 23805	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		hmph 23805	. 2 ⊢ (𝐾 ≃ 𝐿 ↔ (𝐾Homeo𝐿) ≠ ∅)
3		n0 4376	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
4		n0 4376	. . 3 ⊢ ((𝐾Homeo𝐿) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿))
5		exdistrv 1955	. . . 4 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) ↔ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)))
6		hmeoco 23801	. . . . . 6 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → (𝑔 ∘ 𝑓) ∈ (𝐽Homeo𝐿))
7		hmphi 23806	. . . . . 6 ⊢ ((𝑔 ∘ 𝑓) ∈ (𝐽Homeo𝐿) → 𝐽 ≃ 𝐿)
8	6, 7	syl 17	. . . . 5 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽 ≃ 𝐿)
9	8	exlimivv 1931	. . . 4 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽 ≃ 𝐿)
10	5, 9	sylbir 235	. . 3 ⊢ ((∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽 ≃ 𝐿)
11	3, 4, 10	syl2anb 597	. 2 ⊢ (((𝐽Homeo𝐾) ≠ ∅ ∧ (𝐾Homeo𝐿) ≠ ∅) → 𝐽 ≃ 𝐿)
12	1, 2, 11	syl2anb 597	1 ⊢ ((𝐽 ≃ 𝐾 ∧ 𝐾 ≃ 𝐿) → 𝐽 ≃ 𝐿)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946  ∅c0 4352   class class class wbr 5166   ∘ ccom 5704  (class class class)co 7448  Homeochmeo 23782   ≃ chmph 23783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-1o 8522  df-map 8886  df-top 22921  df-topon 22938  df-cn 23256  df-hmeo 23784  df-hmph 23785

This theorem is referenced by:  hmpher  23813  xrhmph  24997