Theorem mapval 8848

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 8846	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})
4	1, 2, 3	mp2an 691	1 ⊢ (𝐴 ↑m 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}

Syntax hints:   = wceq 1534   ∈ wcel 2099  {cab 2704  Vcvv 3469  ⟶wf 6538  (class class class)co 7414   ↑_m cmap 8836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8838

This theorem is referenced by:  0map0sn0  8895  maprnin  32497  poimirlem4  37032  poimirlem9  37037  poimirlem26  37054  poimirlem27  37055  poimirlem28  37056  poimirlem32  37060  lautset  39492  pautsetN  39508  tendoset  40169