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Theorem mapval 8780
Description: The value of set exponentiation (inference version). (𝐴m 𝐵) is the set of all functions that map from 𝐵 to 𝐴. Definition 10.24 of [Kunen] p. 24. (Contributed by NM, 8-Dec-2003.)
Hypotheses
Ref Expression
mapval.1 𝐴 ∈ V
mapval.2 𝐵 ∈ V
Assertion
Ref Expression
mapval (𝐴m 𝐵) = {𝑓𝑓:𝐵𝐴}
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem mapval
StepHypRef Expression
1 mapval.1 . 2 𝐴 ∈ V
2 mapval.2 . 2 𝐵 ∈ V
3 mapvalg 8778 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴m 𝐵) = {𝑓𝑓:𝐵𝐴})
41, 2, 3mp2an 691 1 (𝐴m 𝐵) = {𝑓𝑓:𝐵𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  {cab 2710  Vcvv 3444  wf 6493  (class class class)co 7358  m cmap 8768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770
This theorem is referenced by:  0map0sn0  8826  maprnin  31695  poimirlem4  36128  poimirlem9  36133  poimirlem26  36150  poimirlem27  36151  poimirlem28  36152  poimirlem32  36156  lautset  38591  pautsetN  38607  tendoset  39268
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