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Theorem map0e 8873
Description: Set exponentiation with an empty exponent (ordinal number 0) is ordinal number 1. Exercise 4.42(a) of [Mendelson] p. 255. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 14-Jul-2022.)
Assertion
Ref Expression
map0e (𝐴𝑉 → (𝐴m ∅) = 1o)

Proof of Theorem map0e
StepHypRef Expression
1 mapdm0 8833 . 2 (𝐴𝑉 → (𝐴m ∅) = {∅})
2 df1o2 8469 . 2 1o = {∅}
31, 2eqtr4di 2782 1 (𝐴𝑉 → (𝐴m ∅) = 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  c0 4315  {csn 4621  (class class class)co 7402  1oc1o 8455  m cmap 8817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1o 8462  df-map 8819
This theorem is referenced by:  fseqenlem1  10016  infmap2  10210  pwcfsdom  10575  cfpwsdom  10576  mat0dimbas0  22312  mavmul0  22398  mavmul0g  22399  cramer0  22536  poimirlem28  37019  pwslnmlem0  42383  lincval0  47344  lco0  47356  linds0  47394
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