Theorem 2nd0 7938

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 7934	. 2 ⊢ (2nd ‘∅) = ∪ ran {∅}
2		dmsn0 6162	. . . 4 ⊢ dom {∅} = ∅
3		dm0rn0 5871	. . . 4 ⊢ (dom {∅} = ∅ ↔ ran {∅} = ∅)
4	2, 3	mpbi 230	. . 3 ⊢ ran {∅} = ∅
5	4	unieqi 4873	. 2 ⊢ ∪ ran {∅} = ∪ ∅
6		uni0 4889	. 2 ⊢ ∪ ∅ = ∅
7	1, 5, 6	3eqtri 2756	1 ⊢ (2nd ‘∅) = ∅

Syntax hints:   = wceq 1540  ∅c0 4286  {csn 4579  ∪ cuni 4861  dom cdm 5623  ran crn 5624  ‘cfv 6486  2^nd c2nd 7930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6442  df-fun 6488  df-fv 6494  df-2nd 7932

This theorem is referenced by:  smfval  30567  fucofvalne  49311  reldmprcof2  49368