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Theorem hmeofn 21930
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 21928 . 2 Homeo = (𝑗 ∈ Top, 𝑘 ∈ Top ↦ {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)})
2 ovex 6936 . . 3 (𝑗 Cn 𝑘) ∈ V
32rabex 5036 . 2 {𝑓 ∈ (𝑗 Cn 𝑘) ∣ 𝑓 ∈ (𝑘 Cn 𝑗)} ∈ V
41, 3fnmpt2i 7501 1 Homeo Fn (Top × Top)
Colors of variables: wff setvar class
Syntax hints:  wcel 2166  {crab 3120   × cxp 5339  ccnv 5340   Fn wfn 6117  (class class class)co 6904  Topctop 21067   Cn ccn 21398  Homeochmeo 21926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-fv 6130  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-1st 7427  df-2nd 7428  df-hmeo 21928
This theorem is referenced by:  hmph  21949  hmphtop  21951  hmpher  21957
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