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Theorem map0e 7010
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 4299 . . . 4  |-  (/)  e.  _V
2 elmapg 6990 . . . 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 5523 . . . . . 6  |-  ( f  Fn  (/)  <->  f  =  (/) )
54anbi1i 677 . . . . 5  |-  ( ( f  Fn  (/)  /\  ran  f  C_  A )  <->  ( f  =  (/)  /\  ran  f  C_  A ) )
6 df-f 5417 . . . . 5  |-  ( f : (/) --> A  <->  ( f  Fn  (/)  /\  ran  f  C_  A ) )
7 rneq 5054 . . . . . . . 8  |-  ( f  =  (/)  ->  ran  f  =  ran  (/) )
8 rn0 5086 . . . . . . . 8  |-  ran  (/)  =  (/)
97, 8syl6eq 2452 . . . . . . 7  |-  ( f  =  (/)  ->  ran  f  =  (/) )
10 0ss 3616 . . . . . . 7  |-  (/)  C_  A
119, 10syl6eqss 3358 . . . . . 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 6702 . . . 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 2402 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 1721   _Vcvv 2916    C_ wss 3280   (/)c0 3588   ran crn 4838    Fn wfn 5408   -->wf 5409  (class class class)co 6040   1oc1o 6676    ^m cmap 6977
This theorem is referenced by:  fseqenlem1  7861  infmap2  8054  pwcfsdom  8414  cfpwsdom  8415  hashmap  11653  pwslnmlem0  27061
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1o 6683  df-map 6979
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