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Theorem map0e 6805
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 4150 . . . 4  |-  (/)  e.  _V
2 elmapg 6785 . . . 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 5363 . . . . . 6  |-  ( f  Fn  (/)  <->  f  =  (/) )
54anbi1i 676 . . . . 5  |-  ( ( f  Fn  (/)  /\  ran  f  C_  A )  <->  ( f  =  (/)  /\  ran  f  C_  A ) )
6 df-f 5259 . . . . 5  |-  ( f : (/) --> A  <->  ( f  Fn  (/)  /\  ran  f  C_  A ) )
7 0ss 3483 . . . . . . 7  |-  (/)  C_  A
8 rneq 4904 . . . . . . . . 9  |-  ( f  =  (/)  ->  ran  f  =  ran  (/) )
9 rn0 4936 . . . . . . . . 9  |-  ran  (/)  =  (/)
108, 9syl6eq 2331 . . . . . . . 8  |-  ( f  =  (/)  ->  ran  f  =  (/) )
1110sseq1d 3205 . . . . . . 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 6498 . . . 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 2281 1  |-  ( A  e.  V  ->  ( A  ^m  (/) )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ran crn 4690    Fn wfn 5250   -->wf 5251  (class class class)co 5858   1oc1o 6472    ^m cmap 6772
This theorem is referenced by:  fseqenlem1  7651  infmap2  7844  pwcfsdom  8205  cfpwsdom  8206  hashmap  11387  empklst  26009  pwslnmlem0  27193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1o 6479  df-map 6774
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