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Theorem 2nd0 7217
 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 7213 . 2 (2nd ‘∅) = ran {∅}
2 dmsn0 5637 . . . 4 dom {∅} = ∅
3 dm0rn0 5374 . . . 4 (dom {∅} = ∅ ↔ ran {∅} = ∅)
42, 3mpbi 220 . . 3 ran {∅} = ∅
54unieqi 4477 . 2 ran {∅} =
6 uni0 4497 . 2 ∅ = ∅
71, 5, 63eqtri 2677 1 (2nd ‘∅) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1523  ∅c0 3948  {csn 4210  ∪ cuni 4468  dom cdm 5143  ran crn 5144  ‘cfv 5926  2nd c2nd 7209 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-2nd 7211 This theorem is referenced by:  smfval  27588
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