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Theorem hmeofval 21466
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 6614 . . . 4 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
2 oveq12 6614 . . . . . 6 ((𝑘 = 𝐾𝑗 = 𝐽) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
32ancoms 469 . . . . 5 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
43eleq2d 2689 . . . 4 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑓 ∈ (𝑘 Cn 𝑗) ↔ 𝑓 ∈ (𝐾 Cn 𝐽)))
51, 4rabeqbidv 3186 . . 3 ((𝑗 = 𝐽𝑘 = 𝐾) → {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)} = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
6 df-hmeo 21463 . . 3 Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)})
7 ovex 6633 . . . 4 (𝐽 Cn 𝐾) ∈ V
87rabex 4778 . . 3 {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} ∈ V
95, 6, 8ovmpt2a 6745 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
106mpt2ndm0 6829 . . 3 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = ∅)
11 cntop1 20949 . . . . . . . 8 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
12 cntop2 20950 . . . . . . . 8 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
1311, 12jca 554 . . . . . . 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 151 . . . . 5 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → ¬ 𝑓 ∈ (𝐾 Cn 𝐽)))
1615ralrimiv 2964 . . . 4 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ 𝑓 ∈ (𝐾 Cn 𝐽))
17 rabeq0 3936 . . . 4 ({𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} = ∅ ↔ ∀𝑓 ∈ (𝐽 Cn 𝐾) ¬ 𝑓 ∈ (𝐾 Cn 𝐽))
1816, 17sylibr 224 . . 3 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} = ∅)
1910, 18eqtr4d 2663 . 2 (¬ (𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
209, 19pm2.61i 176 1 (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1992  wral 2912  {crab 2916  c0 3896  ccnv 5078  (class class class)co 6605  Topctop 20612   Cn ccn 20933  Homeochmeo 21461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-top 20616  df-topon 20618  df-cn 20936  df-hmeo 21463
This theorem is referenced by:  ishmeo  21467
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