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Theorem mapval 7756
 Description: The value of set exponentiation (inference version). (𝐴 ↑𝑚 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 7754 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴})
41, 2, 3mp2an 704 1 (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173  ⟶wf 5800  (class class class)co 6549   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746 This theorem is referenced by:  maprnin  28894  poimirlem4  32583  poimirlem9  32588  poimirlem26  32605  poimirlem27  32606  poimirlem28  32607  poimirlem32  32611  lautset  34386  pautsetN  34402  tendoset  35065
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