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Theorem fnmap 6633
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 6628 . 2 𝑚 = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓𝑓:𝑦𝑥})
2 vex 2733 . . 3 𝑦 ∈ V
3 vex 2733 . . 3 𝑥 ∈ V
4 mapex 6632 . . 3 ((𝑦 ∈ V ∧ 𝑥 ∈ V) → {𝑓𝑓:𝑦𝑥} ∈ V)
52, 3, 4mp2an 424 . 2 {𝑓𝑓:𝑦𝑥} ∈ V
61, 5fnmpoi 6183 1 𝑚 Fn (V × V)
Colors of variables: wff set class
Syntax hints:  wcel 2141  {cab 2156  Vcvv 2730   × cxp 4609   Fn wfn 5193  wf 5194  𝑚 cmap 6626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628
This theorem is referenced by:  mapsnen  6789  map1  6790  mapen  6824  mapdom1g  6825  mapxpen  6826  xpmapenlem  6827  hashfacen  10771  omctfn  12398  ismhm  12685  cnfval  12988  cnpfval  12989  cnpval  12992  ismet  13138  isxmet  13139  xmetunirn  13152
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