ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fnmap GIF version

Theorem fnmap 6657
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 6652 . 2 𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑦𝑥})
2 vex 2742 . . 3 𝑦 ∈ V
3 vex 2742 . . 3 𝑥 ∈ V
4 mapex 6656 . . 3 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → {𝑓𝑓:𝑦𝑥} ∈ V)
52, 3, 4mp2an 426 . 2 {𝑓𝑓:𝑦𝑥} ∈ V
61, 5fnmpoi 6207 1 𝑚 Fn (V × V)
Colors of variables: wff set class
Syntax hints:  wcel 2148  {cab 2163  Vcvv 2739   × cxp 4626   Fn wfn 5213  wf 5214  𝑚 cmap 6650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-map 6652
This theorem is referenced by:  mapsnen  6813  map1  6814  mapen  6848  mapdom1g  6849  mapxpen  6850  xpmapenlem  6851  hashfacen  10818  omctfn  12446  ismhm  12858  cnfval  13779  cnpfval  13780  cnpval  13783  ismet  13929  isxmet  13930  xmetunirn  13943
  Copyright terms: Public domain W3C validator