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Theorem map0e 6773
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
StepHypRef Expression
1 0ex 4124 . . . 4  |-  (/)  e.  _V
2 elmapg 6753 . . . 4  |-  ( ( A  e.  V  /\  (/) 
e.  _V )  ->  (
f  e.  ( A  ^m  (/) )  <->  f : (/) --> A ) )
31, 2mpan2 655 . . 3  |-  ( A  e.  V  ->  (
f  e.  ( A  ^m  (/) )  <->  f : (/) --> A ) )
4 fn0 5301 . . . . . 6  |-  ( f  Fn  (/)  <->  f  =  (/) )
54anbi1i 679 . . . . 5  |-  ( ( f  Fn  (/)  /\  ran  f  C_  A )  <->  ( f  =  (/)  /\  ran  f  C_  A ) )
6 df-f 4685 . . . . 5  |-  ( f : (/) --> A  <->  ( f  Fn  (/)  /\  ran  f  C_  A ) )
7 0ss 3458 . . . . . . 7  |-  (/)  C_  A
8 rneq 4892 . . . . . . . . 9  |-  ( f  =  (/)  ->  ran  f  =  ran  (/) )
9 rn0 4924 . . . . . . . . 9  |-  ran  (/)  =  (/)
108, 9syl6eq 2306 . . . . . . . 8  |-  ( f  =  (/)  ->  ran  f  =  (/) )
1110sseq1d 3180 . . . . . . 7  |-  ( f  =  (/)  ->  ( ran  f  C_  A  <->  (/)  C_  A
) )
127, 11mpbiri 226 . . . . . 6  |-  ( f  =  (/)  ->  ran  f  C_  A )
1312pm4.71i 616 . . . . 5  |-  ( f  =  (/)  <->  ( f  =  (/)  /\  ran  f  C_  A ) )
145, 6, 133bitr4i 270 . . . 4  |-  ( f : (/) --> A  <->  f  =  (/) )
15 el1o 6466 . . . 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 2256 1  |-  ( A  e.  V  ->  ( A  ^m  (/) )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2763    C_ wss 3127   (/)c0 3430   ran crn 4662    Fn wfn 4668   -->wf 4669  (class class class)co 5792   1oc1o 6440    ^m cmap 6740
This theorem is referenced by:  fseqenlem1  7619  infmap2  7812  pwcfsdom  8173  cfpwsdom  8174  hashmap  11352  empklst  25376  pwslnmlem0  26560
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-id 4281  df-suc 4370  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1o 6447  df-map 6742
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