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Theorem 2nd0 6383
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 6381 . 2  |-  ( 2nd `  (/) )  =  U. ran  { (/) }
2 dmsn0 5366 . . . 4  |-  dom  { (/)
}  =  (/)
3 dm0rn0 5115 . . . 4  |-  ( dom 
{ (/) }  =  (/)  <->  ran  {
(/) }  =  (/) )
42, 3mpbi 201 . . 3  |-  ran  { (/)
}  =  (/)
54unieqi 4049 . 2  |-  U. ran  {
(/) }  =  U. (/)
6 uni0 4066 . 2  |-  U. (/)  =  (/)
71, 5, 63eqtri 2466 1  |-  ( 2nd `  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1653   (/)c0 3613   {csn 3838   U.cuni 4039   dom cdm 4907   ran crn 4908   ` cfv 5483   2ndc2nd 6377
This theorem is referenced by:  smfval  22115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-iota 5447  df-fun 5485  df-fv 5491  df-2nd 6379
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