Theorem op2nda 5151

Description: Extract the second member of an ordered pair. (See op1sta 5148 to extract the first member and op2ndb 5150 for an alternate version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Hypotheses

Ref	Expression
cnvsn.1	|- A e. _V
cnvsn.2	|- B e. _V

Assertion

Ref	Expression
op2nda	|- U. ran { <. A , B >. } = B

Proof of Theorem op2nda

Step	Hyp	Ref	Expression
1		cnvsn.1	. . . 4 |- A e. _V
2	1	rnsnop 5147	. . 3 |- ran { <. A , B >. } = { B }
3	2	unieqi 3846	. 2 |- U. ran { <. A , B >. } = U. { B }
4		cnvsn.2	. . 3 |- B e. _V
5	4	unisn 3852	. 2 |- U. { B } = B
6	3, 5	eqtri 2214	1 |- U. ran { <. A , B >. } = B

Syntax hints:   = wceq 1364   e. wcel 2164   _Vcvv 2760   {csn 3619   <.cop 3622   U.cuni 3836   ran crn 4661

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-dm 4670  df-rn 4671

This theorem is referenced by:  elxp4  5154  elxp5  5155  op2nd  6202  fo2nd  6213  f2ndres  6215  ixpsnf1o  6792  xpassen  6886  xpdom2  6887