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Theorem fnmap 6479
 Description: Set exponentiation has a universal domain. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
fnmap 𝑚 Fn (V × V)

Proof of Theorem fnmap
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-map 6474 . 2 𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑦𝑥})
2 vex 2644 . . 3 𝑦 ∈ V
3 vex 2644 . . 3 𝑥 ∈ V
4 mapex 6478 . . 3 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → {𝑓𝑓:𝑦𝑥} ∈ V)
52, 3, 4mp2an 420 . 2 {𝑓𝑓:𝑦𝑥} ∈ V
61, 5fnmpoi 6032 1 𝑚 Fn (V × V)
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1448  {cab 2086  Vcvv 2641   × cxp 4475   Fn wfn 5054  ⟶wf 5055   ↑𝑚 cmap 6472 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293 This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-map 6474 This theorem is referenced by:  mapsnen  6635  map1  6636  mapen  6669  mapdom1g  6670  mapxpen  6671  xpmapenlem  6672  hashfacen  10420  cnfval  12145  cnpfval  12146  cnpval  12148  ismet  12272  isxmet  12273  xmetunirn  12286
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