Theorem 2nd0 6317

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		0ex 4221	. . 3 |- (/) e. _V
2		2ndvalg 6315	. . 3 |- ( (/) e. _V -> ( 2nd ` (/) ) = U. ran { (/) } )
3	1, 2	ax-mp 5	. 2 |- ( 2nd ` (/) ) = U. ran { (/) }
4		dmsn0 5211	. . . 4 |- dom { (/) } = (/)
5		dm0rn0 4954	. . . 4 |- ( dom { (/) } = (/) <-> ran { (/) } = (/) )
6	4, 5	mpbi 145	. . 3 |- ran { (/) } = (/)
7	6	unieqi 3908	. 2 |- U. ran { (/) } = U. (/)
8		uni0 3925	. 2 |- U. (/) = (/)
9	3, 7, 8	3eqtri 2256	1 |- ( 2nd ` (/) ) = (/)

Syntax hints:   = wceq 1398   e. wcel 2202   _Vcvv 2803   (/)c0 3496   {csn 3673   U.cuni 3898   dom cdm 4731   ran crn 4732   ` cfv 5333   2ndc2nd 6311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fv 5341  df-2nd 6313

This theorem is referenced by: (None)