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Theorem map0e 7018
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 4307 . . . 4  |-  (/)  e.  _V
2 elmapg 6998 . . . 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 5531 . . . . . 6  |-  ( f  Fn  (/)  <->  f  =  (/) )
54anbi1i 677 . . . . 5  |-  ( ( f  Fn  (/)  /\  ran  f  C_  A )  <->  ( f  =  (/)  /\  ran  f  C_  A ) )
6 df-f 5425 . . . . 5  |-  ( f : (/) --> A  <->  ( f  Fn  (/)  /\  ran  f  C_  A ) )
7 rneq 5062 . . . . . . . 8  |-  ( f  =  (/)  ->  ran  f  =  ran  (/) )
8 rn0 5094 . . . . . . . 8  |-  ran  (/)  =  (/)
97, 8syl6eq 2460 . . . . . . 7  |-  ( f  =  (/)  ->  ran  f  =  (/) )
10 0ss 3624 . . . . . . 7  |-  (/)  C_  A
119, 10syl6eqss 3366 . . . . . 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 6710 . . . 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 2410 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 2924    C_ wss 3288   (/)c0 3596   ran crn 4846    Fn wfn 5416   -->wf 5417  (class class class)co 6048   1oc1o 6684    ^m cmap 6985
This theorem is referenced by:  fseqenlem1  7869  infmap2  8062  pwcfsdom  8422  cfpwsdom  8423  hashmap  11661  pwslnmlem0  27069
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-suc 4555  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1o 6691  df-map 6987
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