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Theorem ishmeo 23969
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 5861 . . 3 (𝑓 = 𝐹𝑓 = 𝐹)
21eleq1d 2850 . 2 (𝑓 = 𝐹 → (𝑓 ∈ (𝐾 Cn 𝐽) ↔ 𝐹 ∈ (𝐾 Cn 𝐽)))
3 hmeofval 23968 . 2 (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)}
42, 3elrab2 3656 1 (𝐹 ∈ (𝐽Homeo𝐾) ↔ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐽)))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wb 209  wa 401   = wceq 1570  wcel 2146  ccnv 5662  (class class class)co 7419   Cn ccn 23433  Homeochmeo 23963
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8832  df-top 23103  df-topon 23120  df-cn 23436  df-hmeo 23965
This theorem is used by:  hmeocn  23970  hmeocnvcn  23971  hmeocnv  23972  hmeores  23981  hmeoco  23982  idhmeo  23983  indishmph  24008  cmphaushmeo  24010  ordthmeo  24012  txhmeo  24013  txswaphmeo  24015  pt1hmeo  24016  ptunhmeo  24018  xkohmeo  24025  qtopf1  24026  qtophmeo  24027  grpinvhmeo  24296  tgplacthmeo  24313  cncfcnvcn  25137  icchmeo  25153  cnrehmeo  25165  cnheiborlem  25166  ismtyhmeo  38516
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