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Theorem map0e 6989
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 4282 . . . 4  |-  (/)  e.  _V
2 elmapg 6969 . . . 4  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  (
f  e.  ( A  ^m  (/) )  <->  f : (/) --> A ) )
31, 2mpan2 653 . . 3  |-  ( A  e.  V  ->  (
f  e.  ( A  ^m  (/) )  <->  f : (/) --> A ) )
4 fn0 5506 . . . . . 6  |-  ( f  Fn  (/)  <->  f  =  (/) )
54anbi1i 677 . . . . 5  |-  ( ( f  Fn  (/)  /\  ran  f  C_  A )  <->  ( f  =  (/)  /\  ran  f  C_  A ) )
6 df-f 5400 . . . . 5  |-  ( f : (/) --> A  <->  ( f  Fn  (/)  /\  ran  f  C_  A ) )
7 rneq 5037 . . . . . . . 8  |-  ( f  =  (/)  ->  ran  f  =  ran  (/) )
8 rn0 5069 . . . . . . . 8  |-  ran  (/)  =  (/)
97, 8syl6eq 2437 . . . . . . 7  |-  ( f  =  (/)  ->  ran  f  =  (/) )
10 0ss 3601 . . . . . . 7  |-  (/)  C_  A
119, 10syl6eqss 3343 . . . . . 6  |-  ( f  =  (/)  ->  ran  f  C_  A )
1211pm4.71i 614 . . . . 5  |-  ( f  =  (/)  <->  ( f  =  (/)  /\  ran  f  C_  A ) )
135, 6, 123bitr4i 269 . . . 4  |-  ( f : (/) --> A  <->  f  =  (/) )
14 el1o 6681 . . . 4  |-  ( f  e.  1o  <->  f  =  (/) )
1513, 14bitr4i 244 . . 3  |-  ( f : (/) --> A  <->  f  e.  1o )
163, 15syl6bb 253 . 2  |-  ( A  e.  V  ->  (
f  e.  ( A  ^m  (/) )  <->  f  e.  1o ) )
1716eqrdv 2387 1  |-  ( A  e.  V  ->  ( A  ^m  (/) )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2901    C_ wss 3265   (/)c0 3573   ran crn 4821    Fn wfn 5391   -->wf 5392  (class class class)co 6022   1oc1o 6655    ^m cmap 6956
This theorem is referenced by:  fseqenlem1  7840  infmap2  8033  pwcfsdom  8393  cfpwsdom  8394  hashmap  11627  pwslnmlem0  26864
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-suc 4530  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1o 6662  df-map 6958
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