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Theorem map0e 8823
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 8783 . 2 (𝐴𝑉 → (𝐴m ∅) = {∅})
2 df1o2 8420 . 2 1o = {∅}
31, 2eqtr4di 2791 1 (𝐴𝑉 → (𝐴m ∅) = 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  c0 4283  {csn 4587  (class class class)co 7358  1oc1o 8406  m cmap 8768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1o 8413  df-map 8770
This theorem is referenced by:  fseqenlem1  9965  infmap2  10159  pwcfsdom  10524  cfpwsdom  10525  mat0dimbas0  21831  mavmul0  21917  mavmul0g  21918  cramer0  22055  poimirlem28  36152  pwslnmlem0  41461  lincval0  46582  lco0  46594  linds0  46632
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