Theorem 2nd0 6139

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 4127	. . 3 |- (/) e. _V
2		2ndvalg 6137	. . 3 |- ( (/) e. _V -> ( 2nd ` (/) ) = U. ran { (/) } )
3	1, 2	ax-mp 5	. 2 |- ( 2nd ` (/) ) = U. ran { (/) }
4		dmsn0 5091	. . . 4 |- dom { (/) } = (/)
5		dm0rn0 4839	. . . 4 |- ( dom { (/) } = (/) <-> ran { (/) } = (/) )
6	4, 5	mpbi 145	. . 3 |- ran { (/) } = (/)
7	6	unieqi 3817	. 2 |- U. ran { (/) } = U. (/)
8		uni0 3834	. 2 |- U. (/) = (/)
9	3, 7, 8	3eqtri 2202	1 |- ( 2nd ` (/) ) = (/)

Syntax hints:   = wceq 1353   e. wcel 2148   _Vcvv 2737   (/)c0 3422   {csn 3591   U.cuni 3807   dom cdm 4622   ran crn 4623   ` cfv 5211   2ndc2nd 6133

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fv 5219  df-2nd 6135

This theorem is referenced by: (None)