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Theorem hmeofn 22368
 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 df-hmeo 22366 . 2 Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)})
2 ovex 7182 . . 3 (𝑗 Cn 𝑘) ∈ V
32rabex 5221 . 2 {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)} ∈ V
41, 3fnmpoi 7763 1 Homeo Fn (Top × Top)
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2115  {crab 3137   × cxp 5540  ◡ccnv 5541   Fn wfn 6338  (class class class)co 7149  Topctop 21504   Cn ccn 21835  Homeochmeo 22364 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-hmeo 22366 This theorem is referenced by:  hmph  22387  hmphtop  22389  hmpher  22395
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