Theorem mapval 7869

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 7867	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ↑𝑚 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴})
4	1, 2, 3	mp2an 708	1 ⊢ (𝐴 ↑𝑚 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}

Syntax hints:   = wceq 1483   ∈ wcel 1990  {cab 2608  Vcvv 3200  ⟶wf 5884  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859

This theorem is referenced by:  maprnin  29506  poimirlem4  33413  poimirlem9  33418  poimirlem26  33435  poimirlem27  33436  poimirlem28  33437  poimirlem32  33441  lautset  35368  pautsetN  35384  tendoset  36047