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Theorem 2nd0 6780
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 6776 . 2  |-  ( 2nd `  (/) )  =  U. ran  { (/) }
2 dmsn0 5458 . . . 4  |-  dom  { (/)
}  =  (/)
3 dm0rn0 5208 . . . 4  |-  ( dom 
{ (/) }  =  (/)  <->  ran  {
(/) }  =  (/) )
42, 3mpbi 208 . . 3  |-  ran  { (/)
}  =  (/)
54unieqi 4244 . 2  |-  U. ran  {
(/) }  =  U. (/)
6 uni0 4262 . 2  |-  U. (/)  =  (/)
71, 5, 63eqtri 2487 1  |-  ( 2nd `  (/) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398   (/)c0 3783   {csn 4016   U.cuni 4235   dom cdm 4988   ran crn 4989   ` cfv 5570   2ndc2nd 6772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-2nd 6774
This theorem is referenced by:  smfval  25696
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