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Theorem map0e 8435
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 8410 . 2 (𝐴𝑉 → (𝐴m ∅) = {∅})
2 df1o2 8105 . 2 1o = {∅}
31, 2syl6eqr 2871 1 (𝐴𝑉 → (𝐴m ∅) = 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  c0 4288  {csn 4557  (class class class)co 7145  1oc1o 8084  m cmap 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1o 8091  df-map 8397
This theorem is referenced by:  fseqenlem1  9438  infmap2  9628  pwcfsdom  9993  cfpwsdom  9994  mat0dimbas0  21003  mavmul0  21089  mavmul0g  21090  cramer0  21227  poimirlem28  34801  pwslnmlem0  39569  lincval0  44398  lco0  44410  linds0  44448
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