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Theorem map0e 8882
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 8842 . 2 (𝐴𝑉 → (𝐴m ∅) = {∅})
2 df1o2 8479 . 2 1o = {∅}
31, 2eqtr4di 2789 1 (𝐴𝑉 → (𝐴m ∅) = 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  c0 4322  {csn 4628  (class class class)co 7412  1oc1o 8465  m cmap 8826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1o 8472  df-map 8828
This theorem is referenced by:  fseqenlem1  10025  infmap2  10219  pwcfsdom  10584  cfpwsdom  10585  mat0dimbas0  22288  mavmul0  22374  mavmul0g  22375  cramer0  22512  poimirlem28  36980  pwslnmlem0  42296  lincval0  47258  lco0  47270  linds0  47308
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