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Theorem mapval 8074
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 8072 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴})
41, 2, 3mp2an 683 1 (𝐴𝑚 𝐵) = {𝑓𝑓:𝐵𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  wcel 2155  {cab 2751  Vcvv 3350  wf 6066  (class class class)co 6844  𝑚 cmap 8062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fv 6078  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-map 8064
This theorem is referenced by:  maprnin  29958  poimirlem4  33840  poimirlem9  33845  poimirlem26  33862  poimirlem27  33863  poimirlem28  33864  poimirlem32  33868  lautset  36041  pautsetN  36057  tendoset  36718
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