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Theorem 2nd0 6127
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 6125 . 2  |-  ( 2nd `  (/) )  =  U. ran  { (/) }
2 dmsn0 5140 . . . 4  |-  dom  { (/)
}  =  (/)
3 dm0rn0 4895 . . . 4  |-  ( dom 
{ (/) }  =  (/)  <->  ran  {
(/) }  =  (/) )
42, 3mpbi 199 . . 3  |-  ran  { (/)
}  =  (/)
54unieqi 3837 . 2  |-  U. ran  {
(/) }  =  U. (/)
6 uni0 3854 . 2  |-  U. (/)  =  (/)
71, 5, 63eqtri 2307 1  |-  ( 2nd `  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1623   (/)c0 3455   {csn 3640   U.cuni 3827   dom cdm 4689   ran crn 4690   ` cfv 5255   2ndc2nd 6121
This theorem is referenced by:  smfval  21161  codval  25724  cmpval  25726
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-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-fv 5263  df-2nd 6123
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