Theorem 2nd0 6369

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 4255	. . 3 |- (/) e. _V
2		2ndvalg 6367	. . 3 |- ( (/) e. _V -> ( 2nd ` (/) ) = U. ran { (/) } )
3	1, 2	ax-mp 5	. 2 |- ( 2nd ` (/) ) = U. ran { (/) }
4		dmsn0 5250	. . . 4 |- dom { (/) } = (/)
5		dm0rn0 4993	. . . 4 |- ( dom { (/) } = (/) <-> ran { (/) } = (/) )
6	4, 5	mpbi 145	. . 3 |- ran { (/) } = (/)
7	6	unieqi 3940	. 2 |- U. ran { (/) } = U. (/)
8		uni0 3957	. 2 |- U. (/) = (/)
9	3, 7, 8	3eqtri 2263	1 |- ( 2nd ` (/) ) = (/)

Syntax hints:   = wceq 1402   e. wcel 2209   _Vcvv 2821   (/)c0 3520   {csn 3705   U.cuni 3930   dom cdm 4769   ran crn 4770   ` cfv 5372   2ndc2nd 6363

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fv 5380  df-2nd 6365

This theorem is referenced by: (None)