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Theorem hmeofval 22909
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 7284 . . . 4 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
2 oveq12 7284 . . . . . 6 ((𝑘 = 𝐾𝑗 = 𝐽) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
32ancoms 459 . . . . 5 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
43eleq2d 2824 . . . 4 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑓 ∈ (𝑘 Cn 𝑗) ↔ 𝑓 ∈ (𝐾 Cn 𝐽)))
51, 4rabeqbidv 3420 . . 3 ((𝑗 = 𝐽𝑘 = 𝐾) → {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)} = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
6 df-hmeo 22906 . . 3 Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)})
7 ovex 7308 . . . 4 (𝐽 Cn 𝐾) ∈ V
87rabex 5256 . . 3 {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} ∈ V
95, 6, 8ovmpoa 7428 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
106mpondm0 7510 . . 3 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = ∅)
11 cntop1 22391 . . . . . . . 8 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
12 cntop2 22392 . . . . . . . 8 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
1311, 12jca 512 . . . . . . 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 155 . . . . 5 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → ¬ 𝑓 ∈ (𝐾 Cn 𝐽)))
1615ralrimiv 3102 . . . 4 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ 𝑓 ∈ (𝐾 Cn 𝐽))
17 rabeq0 4318 . . . 4 ({𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} = ∅ ↔ ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ 𝑓 ∈ (𝐾 Cn 𝐽))
1816, 17sylibr 233 . . 3 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} = ∅)
1910, 18eqtr4d 2781 . 2 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
209, 19pm2.61i 182 1 (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1539  wcel 2106  wral 3064  {crab 3068  c0 4256  ccnv 5588  (class class class)co 7275  Topctop 22042   Cn ccn 22375  Homeochmeo 22904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-top 22043  df-topon 22060  df-cn 22378  df-hmeo 22906
This theorem is referenced by:  ishmeo  22910
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