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Theorem ishmeo 22295
Description: The predicate F is a homeomorphism between topology 𝐽 and topology 𝐾. Criterion of [BourbakiTop1] p. I.2. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
ishmeo (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐽)))

Proof of Theorem ishmeo
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 cnveq 5737 . . 3 (𝑓 = 𝐹𝑓 = 𝐹)
21eleq1d 2894 . 2 (𝑓 = 𝐹 → (𝑓 ∈ (𝐾 Cn 𝐽) ↔ 𝐹 ∈ (𝐾 Cn 𝐽)))
3 hmeofval 22294 . 2 (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)}
42, 3elrab2 3680 1 (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  ccnv 5547  (class class class)co 7145   Cn ccn 21760  Homeochmeo 22289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-top 21430  df-topon 21447  df-cn 21763  df-hmeo 22291
This theorem is referenced by:  hmeocn  22296  hmeocnvcn  22297  hmeocnv  22298  hmeores  22307  hmeoco  22308  idhmeo  22309  indishmph  22334  cmphaushmeo  22336  ordthmeo  22338  txhmeo  22339  txswaphmeo  22341  pt1hmeo  22342  ptunhmeo  22344  xkohmeo  22351  qtopf1  22352  qtophmeo  22353  grpinvhmeo  22622  tgplacthmeo  22639  cncfcnvcn  23456  icchmeo  23472  cnrehmeo  23484  cnheiborlem  23485  ismtyhmeo  34964
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