Theorem op2nda 5252

Description: Extract the second member of an ordered pair. (See op1sta 5249 to extract the first member and op2ndb 5251 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 5248	. . 3 |- ran { <. A , B >. } = { B }
3	2	unieqi 3929	. 2 |- U. ran { <. A , B >. } = U. { B }
4		cnvsn.2	. . 3 |- B e. _V
5	4	unisn 3935	. 2 |- U. { B } = B
6	3, 5	eqtri 2255	1 |- U. ran { <. A , B >. } = B

Syntax hints:   = wceq 1398   e. wcel 2205   _Vcvv 2815   {csn 3694   <.cop 3697   U.cuni 3919   ran crn 4755

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 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-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762  df-dm 4764  df-rn 4765

This theorem is referenced by:  elxp4  5255  elxp5  5256  op2nd  6354  fo2nd  6365  f2ndres  6367  ixpsnf1o  6984  xpassen  7094  xpdom2  7095