Theorem 2nd0 6289

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 4210	. . 3 |- (/) e. _V
2		2ndvalg 6287	. . 3 |- ( (/) e. _V -> ( 2nd ` (/) ) = U. ran { (/) } )
3	1, 2	ax-mp 5	. 2 |- ( 2nd ` (/) ) = U. ran { (/) }
4		dmsn0 5195	. . . 4 |- dom { (/) } = (/)
5		dm0rn0 4939	. . . 4 |- ( dom { (/) } = (/) <-> ran { (/) } = (/) )
6	4, 5	mpbi 145	. . 3 |- ran { (/) } = (/)
7	6	unieqi 3897	. 2 |- U. ran { (/) } = U. (/)
8		uni0 3914	. 2 |- U. (/) = (/)
9	3, 7, 8	3eqtri 2254	1 |- ( 2nd ` (/) ) = (/)

Syntax hints:   = wceq 1395   e. wcel 2200   _Vcvv 2799   (/)c0 3491   {csn 3666   U.cuni 3887   dom cdm 4718   ran crn 4719   ` cfv 5317   2ndc2nd 6283

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fv 5325  df-2nd 6285

This theorem is referenced by: (None)