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Theorem ishmeo 21320
Description: The predicate F is a homeomorphism between topology 𝐽 and topology 𝐾. Proposition 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 5206 . . 3 (𝑓 = 𝐹𝑓 = 𝐹)
21eleq1d 2671 . 2 (𝑓 = 𝐹 → (𝑓 ∈ (𝐾 Cn 𝐽) ↔ 𝐹 ∈ (𝐾 Cn 𝐽)))
3 hmeofval 21319 . 2 (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)}
42, 3elrab2 3332 1 (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wcel 1976  ccnv 5027  (class class class)co 6527   Cn ccn 20786  Homeochmeo 21314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-map 7724  df-top 20469  df-topon 20471  df-cn 20789  df-hmeo 21316
This theorem is referenced by:  hmeocn  21321  hmeocnvcn  21322  hmeocnv  21323  hmeores  21332  hmeoco  21333  idhmeo  21334  indishmph  21359  cmphaushmeo  21361  ordthmeo  21363  txhmeo  21364  txswaphmeo  21366  pt1hmeo  21367  ptunhmeo  21369  xkohmeo  21376  qtopf1  21377  qtophmeo  21378  grpinvhmeo  21648  tgplacthmeo  21665  cncfcnvcn  22480  icchmeo  22496  cnrehmeo  22508  cnheiborlem  22509  ismtyhmeo  32598
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