Theorem 2nd0 6339

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 4237	. . 3 |- (/) e. _V
2		2ndvalg 6337	. . 3 |- ( (/) e. _V -> ( 2nd ` (/) ) = U. ran { (/) } )
3	1, 2	ax-mp 5	. 2 |- ( 2nd ` (/) ) = U. ran { (/) }
4		dmsn0 5230	. . . 4 |- dom { (/) } = (/)
5		dm0rn0 4973	. . . 4 |- ( dom { (/) } = (/) <-> ran { (/) } = (/) )
6	4, 5	mpbi 145	. . 3 |- ran { (/) } = (/)
7	6	unieqi 3924	. 2 |- U. ran { (/) } = U. (/)
8		uni0 3941	. 2 |- U. (/) = (/)
9	3, 7, 8	3eqtri 2257	1 |- ( 2nd ` (/) ) = (/)

Syntax hints:   = wceq 1398   e. wcel 2203   _Vcvv 2813   (/)c0 3508   {csn 3689   U.cuni 3914   dom cdm 4749   ran crn 4750   ` cfv 5352   2ndc2nd 6333

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fv 5360  df-2nd 6335

This theorem is referenced by: (None)