Theorem mapval 8878

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	⊢ (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem mapval

Step	Hyp	Ref	Expression
1		mapval.1	. 2 ⊢ 𝐴 ∈ V
2		mapval.2	. 2 ⊢ 𝐵 ∈ V
3		mapvalg 8876	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})
4	1, 2, 3	mp2an 692	1 ⊢ (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}

Syntax hints:   = wceq 1540   ∈ wcel 2108  {cab 2714  Vcvv 3480  ⟶wf 6557  (class class class)co 7431   ↑_m cmap 8866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868

This theorem is referenced by:  0map0sn0  8925  maprnin  32742  poimirlem4  37631  poimirlem9  37636  poimirlem26  37653  poimirlem27  37654  poimirlem28  37655  poimirlem32  37659  lautset  40084  pautsetN  40100  tendoset  40761