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Theorem map0e 7043
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 4331 . . . 4  |-  (/)  e.  _V
2 elmapg 7023 . . . 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 5556 . . . . . 6  |-  ( f  Fn  (/)  <->  f  =  (/) )
54anbi1i 677 . . . . 5  |-  ( ( f  Fn  (/)  /\  ran  f  C_  A )  <->  ( f  =  (/)  /\  ran  f  C_  A ) )
6 df-f 5450 . . . . 5  |-  ( f : (/) --> A  <->  ( f  Fn  (/)  /\  ran  f  C_  A ) )
7 rneq 5087 . . . . . . . 8  |-  ( f  =  (/)  ->  ran  f  =  ran  (/) )
8 rn0 5119 . . . . . . . 8  |-  ran  (/)  =  (/)
97, 8syl6eq 2483 . . . . . . 7  |-  ( f  =  (/)  ->  ran  f  =  (/) )
10 0ss 3648 . . . . . . 7  |-  (/)  C_  A
119, 10syl6eqss 3390 . . . . . 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 6735 . . . 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 2433 1  |-  ( A  e.  V  ->  ( A  ^m  (/) )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   (/)c0 3620   ran crn 4871    Fn wfn 5441   -->wf 5442  (class class class)co 6073   1oc1o 6709    ^m cmap 7010
This theorem is referenced by:  fseqenlem1  7897  infmap2  8090  pwcfsdom  8450  cfpwsdom  8451  hashmap  11690  pwslnmlem0  27161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1o 6716  df-map 7012
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