Theorem 2nd0 7838

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

Step	Hyp	Ref	Expression
1		2ndval 7834	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6112	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5834	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 229	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4852	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4869	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2770	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1539  ∅c0 4256  {csn 4561  ∪ cuni 4839  dom cdm 5589  ran crn 5590  ‘cfv 6433  2^nd c2nd 7830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-2nd 7832

This theorem is referenced by:  smfval  28967