Theorem hmphtr 23748

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 23741	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		hmph 23741	. 2 ⊢ (𝐾 ≃ 𝐿 ↔ (𝐾Homeo𝐿) ≠ ∅)
3		n0 4293	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
4		n0 4293	. . 3 ⊢ ((𝐾Homeo𝐿) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿))
5		exdistrv 1957	. . . 4 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) ↔ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)))
6		hmeoco 23737	. . . . . 6 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → (𝑔 ∘ 𝑓) ∈ (𝐽Homeo𝐿))
7		hmphi 23742	. . . . . 6 ⊢ ((𝑔 ∘ 𝑓) ∈ (𝐽Homeo𝐿) → 𝐽 ≃ 𝐿)
8	6, 7	syl 17	. . . . 5 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → 𝐽 ≃ 𝐿)
9	8	exlimivv 1934	. . . 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 599	. 2 ⊢ (((𝐽Homeo𝐾) ≠ ∅ ∧ (𝐾Homeo𝐿) ≠ ∅) → 𝐽 ≃ 𝐿)
12	1, 2, 11	syl2anb 599	1 ⊢ ((𝐽 ≃ 𝐾 ∧ 𝐾 ≃ 𝐿) → 𝐽 ≃ 𝐿)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∅c0 4273   class class class wbr 5085   ∘ ccom 5635  (class class class)co 7367  Homeochmeo 23718   ≃ chmph 23719

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-1o 8405  df-map 8775  df-top 22859  df-topon 22876  df-cn 23192  df-hmeo 23720  df-hmph 23721

This theorem is referenced by:  hmpher  23749  xrhmph  24914