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Theorem 2nd0 6340
Description: The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
Assertion
Ref Expression
2nd0  |-  ( 2nd `  (/) )  =  (/)

Proof of Theorem 2nd0
StepHypRef Expression
1 2ndval 6338 . 2  |-  ( 2nd `  (/) )  =  U. ran  { (/) }
2 dmsn0 5323 . . . 4  |-  dom  { (/)
}  =  (/)
3 dm0rn0 5072 . . . 4  |-  ( dom 
{ (/) }  =  (/)  <->  ran  {
(/) }  =  (/) )
42, 3mpbi 200 . . 3  |-  ran  { (/)
}  =  (/)
54unieqi 4012 . 2  |-  U. ran  {
(/) }  =  U. (/)
6 uni0 4029 . 2  |-  U. (/)  =  (/)
71, 5, 63eqtri 2454 1  |-  ( 2nd `  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1652   (/)c0 3615   {csn 3801   U.cuni 4002   dom cdm 4864   ran crn 4865   ` cfv 5440   2ndc2nd 6334
This theorem is referenced by:  smfval  22067
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 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
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 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-iota 5404  df-fun 5442  df-fv 5448  df-2nd 6336
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