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Theorem map0e 7476
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

Proof of Theorem map0e
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 0ex 4582 . . . 4
2 elmapg 7452 . . . 4
31, 2mpan2 671 . . 3
4 f0bi 5773 . . . 4
5 el1o 7168 . . . 4
64, 5bitr4i 252 . . 3
73, 6syl6bb 261 . 2
87eqrdv 2454 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  e.wcel 1818   cvv 3109   c0 3784  -->wf 5589  (class class class)co 6296   c1o 7142   cmap 7439
This theorem is referenced by:  fseqenlem1  8426  infmap2  8619  pwcfsdom  8979  cfpwsdom  8980  hashmap  12493  mat0dimbas0  18968  mavmul0  19054  mavmul0g  19055  cramer0  19192  pwslnmlem0  31037  lincval0  33016  lco0  33028  linds0  33066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1o 7149  df-map 7441
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