Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  map0e Unicode version

Theorem map0e 6805
 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 4150 . . . 4
2 elmapg 6785 . . . 4
31, 2mpan2 652 . . 3
4 fn0 5363 . . . . . 6
54anbi1i 676 . . . . 5
6 df-f 5259 . . . . 5
7 0ss 3483 . . . . . . 7
8 rneq 4904 . . . . . . . . 9
9 rn0 4936 . . . . . . . . 9
108, 9syl6eq 2331 . . . . . . . 8
1110sseq1d 3205 . . . . . . 7
127, 11mpbiri 224 . . . . . 6
1312pm4.71i 613 . . . . 5
145, 6, 133bitr4i 268 . . . 4
15 el1o 6498 . . . 4
1614, 15bitr4i 243 . . 3
173, 16syl6bb 252 . 2
1817eqrdv 2281 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1623   wcel 1684  cvv 2788   wss 3152  c0 3455   crn 4690   wfn 5250  wf 5251  (class class class)co 5858  c1o 6472   cmap 6772 This theorem is referenced by:  fseqenlem1  7651  infmap2  7844  pwcfsdom  8205  cfpwsdom  8206  hashmap  11387  empklst  26009  pwslnmlem0  27193 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1o 6479  df-map 6774
 Copyright terms: Public domain W3C validator