Theorem 2nd0 7949

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 7945	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6173	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5879	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 230	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4862	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4878	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2763	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1542  ∅c0 4273  {csn 4567  ∪ cuni 4850  dom cdm 5631  ran crn 5632  ‘cfv 6498  2^nd c2nd 7941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fv 6506  df-2nd 7943

This theorem is referenced by:  smfval  30676  fucofvalne  49800  reldmprcof2  49857