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Theorem map0e 6586
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.)
Assertion
Ref Expression
map0e  |-  ( A  e.  V  ->  ( A  ^m  (/) )  =  1o )

Proof of Theorem map0e
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 0ex 4061 . . . 4  |-  (/)  e.  _V
2 elmapg 6561 . . . 4  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  (
f  e.  ( A  ^m  (/) )  <->  f : (/) --> A ) )
31, 2mpan2 422 . . 3  |-  ( A  e.  V  ->  (
f  e.  ( A  ^m  (/) )  <->  f : (/) --> A ) )
4 f0bi 5321 . . . 4  |-  ( f : (/) --> A  <->  f  =  (/) )
5 el1o 6340 . . . 4  |-  ( f  e.  1o  <->  f  =  (/) )
64, 5bitr4i 186 . . 3  |-  ( f : (/) --> A  <->  f  e.  1o )
73, 6syl6bb 195 . 2  |-  ( A  e.  V  ->  (
f  e.  ( A  ^m  (/) )  <->  f  e.  1o ) )
87eqrdv 2138 1  |-  ( A  e.  V  ->  ( A  ^m  (/) )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1332    e. wcel 1481   _Vcvv 2689   (/)c0 3366   -->wf 5125  (class class class)co 5780   1oc1o 6312    ^m cmap 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-nul 4060  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-nul 3367  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-br 3936  df-opab 3996  df-id 4221  df-suc 4299  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-rn 4556  df-iota 5094  df-fun 5131  df-fn 5132  df-f 5133  df-fv 5137  df-ov 5783  df-oprab 5784  df-mpo 5785  df-1o 6319  df-map 6550
This theorem is referenced by: (None)
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