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Theorem mapval 8831
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 8829 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴m 𝐵) = {𝑓𝑓:𝐵𝐴})
41, 2, 3mp2an 689 1 (𝐴m 𝐵) = {𝑓𝑓:𝐵𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  {cab 2703  Vcvv 3468  wf 6532  (class class class)co 7404  m cmap 8819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821
This theorem is referenced by:  0map0sn0  8878  maprnin  32461  poimirlem4  37003  poimirlem9  37008  poimirlem26  37025  poimirlem27  37026  poimirlem28  37027  poimirlem32  37031  lautset  39464  pautsetN  39480  tendoset  40141
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