Theorem hmphtr 23739

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 23732	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		hmph 23732	. 2 ⊢ (𝐾 ≃ 𝐿 ↔ (𝐾Homeo𝐿) ≠ ∅)
3		n0 4307	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
4		n0 4307	. . 3 ⊢ ((𝐾Homeo𝐿) ≠ ∅ ↔ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿))
5		exdistrv 1957	. . . 4 ⊢ (∃𝑓∃𝑔(𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) ↔ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) ∧ ∃𝑔 𝑔 ∈ (𝐾Homeo𝐿)))
6		hmeoco 23728	. . . . . 6 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑔 ∈ (𝐾Homeo𝐿)) → (𝑔 ∘ 𝑓) ∈ (𝐽Homeo𝐿))
7		hmphi 23733	. . . . . 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 2933  ∅c0 4287   class class class wbr 5100   ∘ ccom 5636  (class class class)co 7368  Homeochmeo 23709   ≃ chmph 23710

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 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-1o 8407  df-map 8777  df-top 22850  df-topon 22867  df-cn 23183  df-hmeo 23711  df-hmph 23712

This theorem is referenced by:  hmpher  23740  xrhmph  24913