Theorem 2nd0 7811

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 7807	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6101	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5823	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 229	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4849	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4866	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2770	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1539  ∅c0 4253  {csn 4558  ∪ cuni 4836  dom cdm 5580  ran crn 5581  ‘cfv 6418  2^nd c2nd 7803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-2nd 7805

This theorem is referenced by:  smfval  28868