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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

Proof of Theorem map0e
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 0ex 4331 . . . 4
2 elmapg 7023 . . . 4
31, 2mpan2 653 . . 3
4 fn0 5556 . . . . . 6
54anbi1i 677 . . . . 5
6 df-f 5450 . . . . 5
7 rneq 5087 . . . . . . . 8
8 rn0 5119 . . . . . . . 8
97, 8syl6eq 2483 . . . . . . 7
10 0ss 3648 . . . . . . 7
119, 10syl6eqss 3390 . . . . . 6
1211pm4.71i 614 . . . . 5
135, 6, 123bitr4i 269 . . . 4
14 el1o 6735 . . . 4
1513, 14bitr4i 244 . . 3
163, 15syl6bb 253 . 2
1716eqrdv 2433 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cvv 2948   wss 3312  c0 3620   crn 4871   wfn 5441  wf 5442  (class class class)co 6073  c1o 6709   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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