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Theorem fncvm 31578
Description: Lemma for covering maps. (Contributed by Mario Carneiro, 13-Feb-2015.)
Assertion
Ref Expression
fncvm CovMap Fn (Top × Top)

Proof of Theorem fncvm
Dummy variables 𝑗 𝑐 𝑓 𝑥 𝑘 𝑠 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cvm 31577 . 2 CovMap = (𝑐 ∈ Top, 𝑗 ∈ Top ↦ {𝑓 ∈ (𝑐 Cn 𝑗) ∣ ∀𝑥 𝑗𝑘𝑗 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘)))))})
2 ovex 6824 . . 3 (𝑐 Cn 𝑗) ∈ V
32rabex 4947 . 2 {𝑓 ∈ (𝑐 Cn 𝑗) ∣ ∀𝑥 𝑗𝑘𝑗 (𝑥𝑘 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∖ {∅})( 𝑠 = (𝑓𝑘) ∧ ∀𝑢𝑠 (∀𝑣 ∈ (𝑠 ∖ {𝑢})(𝑢𝑣) = ∅ ∧ (𝑓𝑢) ∈ ((𝑐t 𝑢)Homeo(𝑗t 𝑘)))))} ∈ V
41, 3fnmpt2i 7390 1 CovMap Fn (Top × Top)
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  {crab 3065  cdif 3721  cin 3723  c0 4064  𝒫 cpw 4298  {csn 4317   cuni 4575   × cxp 5248  ccnv 5249  cres 5252  cima 5253   Fn wfn 6027  (class class class)co 6794  t crest 16290  Topctop 20919   Cn ccn 21250  Homeochmeo 21778   CovMap ccvm 31576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-1st 7316  df-2nd 7317  df-cvm 31577
This theorem is referenced by:  cvmtop1  31581  cvmtop2  31582
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