Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  hmeofn GIF version

Theorem hmeofn 12544
 Description: The set of homeomorphisms is a function on topologies. (Contributed by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
hmeofn Homeo Fn (Top × Top)

Proof of Theorem hmeofn
Dummy variables 𝑓 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnovex 12438 . . . 4 ((𝑗 ∈ Top ∧ 𝑘 ∈ Top) → (𝑗 Cn 𝑘) ∈ V)
2 rabexg 4080 . . . 4 ((𝑗 Cn 𝑘) ∈ V → {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)} ∈ V)
31, 2syl 14 . . 3 ((𝑗 ∈ Top ∧ 𝑘 ∈ Top) → {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)} ∈ V)
43rgen2a 2490 . 2 𝑗 ∈ Top ∀𝑘 ∈ Top {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)} ∈ V
5 df-hmeo 12543 . . 3 Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)})
65fnmpo 6111 . 2 (∀𝑗 ∈ Top ∀𝑘 ∈ Top {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)} ∈ V → Homeo Fn (Top × Top))
74, 6ax-mp 5 1 Homeo Fn (Top × Top)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ∈ wcel 1481  ∀wral 2417  {crab 2421  Vcvv 2690   × cxp 4548  ◡ccnv 4549   Fn wfn 5129  (class class class)co 5785  Topctop 12237   Cn ccn 12427  Homeochmeo 12542 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4141  ax-un 4365  ax-setind 4462 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4225  df-xp 4556  df-rel 4557  df-cnv 4558  df-co 4559  df-dm 4560  df-rn 4561  df-res 4562  df-ima 4563  df-iota 5099  df-fun 5136  df-fn 5137  df-f 5138  df-fv 5142  df-ov 5788  df-oprab 5789  df-mpo 5790  df-1st 6049  df-2nd 6050  df-map 6555  df-top 12238  df-topon 12251  df-cn 12430  df-hmeo 12543 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator