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Theorem map0e 6800
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 )
Dummy variable  f is distinct from all other variables.

Proof of Theorem map0e
StepHypRef Expression
1 0ex 4151 . . . 4  |-  (/)  e.  _V
2 elmapg 6780 . . . 4  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  (
f  e.  ( A  ^m  (/) )  <->  f : (/) --> A ) )
31, 2mpan2 654 . . 3  |-  ( A  e.  V  ->  (
f  e.  ( A  ^m  (/) )  <->  f : (/) --> A ) )
4 fn0 5328 . . . . . 6  |-  ( f  Fn  (/)  <->  f  =  (/) )
54anbi1i 678 . . . . 5  |-  ( ( f  Fn  (/)  /\  ran  f  C_  A )  <->  ( f  =  (/)  /\  ran  f  C_  A ) )
6 df-f 5225 . . . . 5  |-  ( f : (/) --> A  <->  ( f  Fn  (/)  /\  ran  f  C_  A ) )
7 0ss 3484 . . . . . . 7  |-  (/)  C_  A
8 rneq 4903 . . . . . . . . 9  |-  ( f  =  (/)  ->  ran  f  =  ran  (/) )
9 rn0 4935 . . . . . . . . 9  |-  ran  (/)  =  (/)
108, 9syl6eq 2332 . . . . . . . 8  |-  ( f  =  (/)  ->  ran  f  =  (/) )
1110sseq1d 3206 . . . . . . 7  |-  ( f  =  (/)  ->  ( ran  f  C_  A  <->  (/)  C_  A
) )
127, 11mpbiri 226 . . . . . 6  |-  ( f  =  (/)  ->  ran  f  C_  A )
1312pm4.71i 615 . . . . 5  |-  ( f  =  (/)  <->  ( f  =  (/)  /\  ran  f  C_  A ) )
145, 6, 133bitr4i 270 . . . 4  |-  ( f : (/) --> A  <->  f  =  (/) )
15 el1o 6493 . . . 4  |-  ( f  e.  1o  <->  f  =  (/) )
1614, 15bitr4i 245 . . 3  |-  ( f : (/) --> A  <->  f  e.  1o )
173, 16syl6bb 254 . 2  |-  ( A  e.  V  ->  (
f  e.  ( A  ^m  (/) )  <->  f  e.  1o ) )
1817eqrdv 2282 1  |-  ( A  e.  V  ->  ( A  ^m  (/) )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   _Vcvv 2789    C_ wss 3153   (/)c0 3456   ran crn 4689    Fn wfn 5216   -->wf 5217  (class class class)co 5819   1oc1o 6467    ^m cmap 6767
This theorem is referenced by:  fseqenlem1  7646  infmap2  7839  pwcfsdom  8200  cfpwsdom  8201  hashmap  11381  empklst  25408  pwslnmlem0  26592
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1o 6474  df-map 6769
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