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Theorem 2nd0 6802
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 6798 . 2  |-  ( 2nd `  (/) )  =  U. ran  { (/) }
2 dmsn0 5481 . . . 4  |-  dom  { (/)
}  =  (/)
3 dm0rn0 5225 . . . 4  |-  ( dom 
{ (/) }  =  (/)  <->  ran  {
(/) }  =  (/) )
42, 3mpbi 208 . . 3  |-  ran  { (/)
}  =  (/)
54unieqi 4260 . 2  |-  U. ran  {
(/) }  =  U. (/)
6 uni0 4278 . 2  |-  U. (/)  =  (/)
71, 5, 63eqtri 2500 1  |-  ( 2nd `  (/) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   (/)c0 3790   {csn 4033   U.cuni 4251   dom cdm 5005   ran crn 5006   ` cfv 5594   2ndc2nd 6794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fv 5602  df-2nd 6796
This theorem is referenced by:  smfval  25321
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