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Theorem 2nd0 7160
 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 7156 . 2 (2nd ‘∅) = ran {∅}
2 dmsn0 5590 . . . 4 dom {∅} = ∅
3 dm0rn0 5331 . . . 4 (dom {∅} = ∅ ↔ ran {∅} = ∅)
42, 3mpbi 220 . . 3 ran {∅} = ∅
54unieqi 4436 . 2 ran {∅} =
6 uni0 4456 . 2 ∅ = ∅
71, 5, 63eqtri 2646 1 (2nd ‘∅) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1481  ∅c0 3907  {csn 4168  ∪ cuni 4427  dom cdm 5104  ran crn 5105  ‘cfv 5876  2nd c2nd 7152 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fv 5884  df-2nd 7154 This theorem is referenced by:  smfval  27430
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