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Theorem map0e 6821
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 4166 . . . 4  |-  (/)  e.  _V
2 elmapg 6801 . . . 4  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  (
f  e.  ( A  ^m  (/) )  <->  f : (/) --> A ) )
31, 2mpan2 652 . . 3  |-  ( A  e.  V  ->  (
f  e.  ( A  ^m  (/) )  <->  f : (/) --> A ) )
4 fn0 5379 . . . . . 6  |-  ( f  Fn  (/)  <->  f  =  (/) )
54anbi1i 676 . . . . 5  |-  ( ( f  Fn  (/)  /\  ran  f  C_  A )  <->  ( f  =  (/)  /\  ran  f  C_  A ) )
6 df-f 5275 . . . . 5  |-  ( f : (/) --> A  <->  ( f  Fn  (/)  /\  ran  f  C_  A ) )
7 0ss 3496 . . . . . . 7  |-  (/)  C_  A
8 rneq 4920 . . . . . . . . 9  |-  ( f  =  (/)  ->  ran  f  =  ran  (/) )
9 rn0 4952 . . . . . . . . 9  |-  ran  (/)  =  (/)
108, 9syl6eq 2344 . . . . . . . 8  |-  ( f  =  (/)  ->  ran  f  =  (/) )
1110sseq1d 3218 . . . . . . 7  |-  ( f  =  (/)  ->  ( ran  f  C_  A  <->  (/)  C_  A
) )
127, 11mpbiri 224 . . . . . 6  |-  ( f  =  (/)  ->  ran  f  C_  A )
1312pm4.71i 613 . . . . 5  |-  ( f  =  (/)  <->  ( f  =  (/)  /\  ran  f  C_  A ) )
145, 6, 133bitr4i 268 . . . 4  |-  ( f : (/) --> A  <->  f  =  (/) )
15 el1o 6514 . . . 4  |-  ( f  e.  1o  <->  f  =  (/) )
1614, 15bitr4i 243 . . 3  |-  ( f : (/) --> A  <->  f  e.  1o )
173, 16syl6bb 252 . 2  |-  ( A  e.  V  ->  (
f  e.  ( A  ^m  (/) )  <->  f  e.  1o ) )
1817eqrdv 2294 1  |-  ( A  e.  V  ->  ( A  ^m  (/) )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ran crn 4706    Fn wfn 5266   -->wf 5267  (class class class)co 5874   1oc1o 6488    ^m cmap 6788
This theorem is referenced by:  fseqenlem1  7667  infmap2  7860  pwcfsdom  8221  cfpwsdom  8222  hashmap  11403  empklst  26112  pwslnmlem0  27296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1o 6495  df-map 6790
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