Theorem mapval 8785

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 8783	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})
4	1, 2, 3	mp2an 693	1 ⊢ (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}

Syntax hints:   = wceq 1542   ∈ wcel 2114  {cab 2714  Vcvv 3429  ⟶wf 6494  (class class class)co 7367   ↑_m cmap 8773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-map 8775

This theorem is referenced by:  0map0sn0  8833  maprnin  32804  poimirlem4  37945  poimirlem9  37950  poimirlem26  37967  poimirlem27  37968  poimirlem28  37969  poimirlem32  37973  lautset  40528  pautsetN  40544  tendoset  41205