MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmeofval Structured version   Visualization version   GIF version

Theorem hmeofval 22502
Description: The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmeofval (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)}
Distinct variable groups:   𝑓,𝐽   𝑓,𝐾

Proof of Theorem hmeofval
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 7173 . . . 4 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
2 oveq12 7173 . . . . . 6 ((𝑘 = 𝐾𝑗 = 𝐽) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
32ancoms 462 . . . . 5 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
43eleq2d 2818 . . . 4 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑓 ∈ (𝑘 Cn 𝑗) ↔ 𝑓 ∈ (𝐾 Cn 𝐽)))
51, 4rabeqbidv 3386 . . 3 ((𝑗 = 𝐽𝑘 = 𝐾) → {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)} = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
6 df-hmeo 22499 . . 3 Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)})
7 ovex 7197 . . . 4 (𝐽 Cn 𝐾) ∈ V
87rabex 5197 . . 3 {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} ∈ V
95, 6, 8ovmpoa 7314 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
106mpondm0 7396 . . 3 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = ∅)
11 cntop1 21984 . . . . . . . 8 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
12 cntop2 21985 . . . . . . . 8 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
1311, 12jca 515 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
1413a1d 25 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → (𝑓 ∈ (𝐾 Cn 𝐽) → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)))
1514con3rr3 158 . . . . 5 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → ¬ 𝑓 ∈ (𝐾 Cn 𝐽)))
1615ralrimiv 3095 . . . 4 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ 𝑓 ∈ (𝐾 Cn 𝐽))
17 rabeq0 4270 . . . 4 ({𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} = ∅ ↔ ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ 𝑓 ∈ (𝐾 Cn 𝐽))
1816, 17sylibr 237 . . 3 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} = ∅)
1910, 18eqtr4d 2776 . 2 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
209, 19pm2.61i 185 1 (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1542  wcel 2113  wral 3053  {crab 3057  c0 4209  ccnv 5518  (class class class)co 7164  Topctop 21637   Cn ccn 21968  Homeochmeo 22497
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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-map 8432  df-top 21638  df-topon 21655  df-cn 21971  df-hmeo 22499
This theorem is referenced by:  ishmeo  22503
  Copyright terms: Public domain W3C validator