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Theorem hmeofvalg 12472
Description: The set of all the homeomorphisms between two topologies. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
hmeofvalg ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
Distinct variable groups:   𝑓,𝐽   𝑓,𝐾

Proof of Theorem hmeofvalg
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnovex 12365 . . 3 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
2 rabexg 4071 . . 3 ((𝐽 Cn 𝐾) ∈ V → {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} ∈ V)
31, 2syl 14 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} ∈ V)
4 oveq12 5783 . . . 4 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑗 Cn 𝑘) = (𝐽 Cn 𝐾))
5 oveq12 5783 . . . . . 6 ((𝑘 = 𝐾𝑗 = 𝐽) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
65ancoms 266 . . . . 5 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑘 Cn 𝑗) = (𝐾 Cn 𝐽))
76eleq2d 2209 . . . 4 ((𝑗 = 𝐽𝑘 = 𝐾) → (𝑓 ∈ (𝑘 Cn 𝑗) ↔ 𝑓 ∈ (𝐾 Cn 𝐽)))
84, 7rabeqbidv 2681 . . 3 ((𝑗 = 𝐽𝑘 = 𝐾) → {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)} = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
9 df-hmeo 12470 . . 3 Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)})
108, 9ovmpoga 5900 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)} ∈ V) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
113, 10mpd3an3 1316 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽Homeo𝐾) = {𝑓 ∈ (𝐽 Cn 𝐾) ∣ 𝑓 ∈ (𝐾 Cn 𝐽)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  {crab 2420  Vcvv 2686  ccnv 4538  (class class class)co 5774  Topctop 12164   Cn ccn 12354  Homeochmeo 12469
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-top 12165  df-topon 12178  df-cn 12357  df-hmeo 12470
This theorem is referenced by:  ishmeo  12473
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