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Theorem ishmeo 22367
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 5712 . . 3 (𝑓 = 𝐹𝑓 = 𝐹)
21eleq1d 2877 . 2 (𝑓 = 𝐹 → (𝑓 ∈ (𝐾 Cn 𝐽) ↔ 𝐹 ∈ (𝐾 Cn 𝐽)))
3 hmeofval 22366 . 2 (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)}
42, 3elrab2 3634 1 (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2112  ccnv 5522  (class class class)co 7139   Cn ccn 21832  Homeochmeo 22361
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-top 21502  df-topon 21519  df-cn 21835  df-hmeo 22363
This theorem is referenced by:  hmeocn  22368  hmeocnvcn  22369  hmeocnv  22370  hmeores  22379  hmeoco  22380  idhmeo  22381  indishmph  22406  cmphaushmeo  22408  ordthmeo  22410  txhmeo  22411  txswaphmeo  22413  pt1hmeo  22414  ptunhmeo  22416  xkohmeo  22423  qtopf1  22424  qtophmeo  22425  grpinvhmeo  22694  tgplacthmeo  22711  cncfcnvcn  23533  icchmeo  23549  cnrehmeo  23561  cnheiborlem  23562  ismtyhmeo  35236
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