Theorem op2nda 5620

Description: Extract the second member of an ordered pair. (See op1sta 5617 to extract the first member, op2ndb 5619 for an alternate version, and op2nd 7177 for the preferred 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 5616	. . 3 |- ran { <. A , B >. } = { B }
3	2	unieqi 4445	. 2 |- U. ran { <. A , B >. } = U. { B }
4		cnvsn.2	. . 3 |- B e. _V
5	4	unisn 4451	. 2 |- U. { B } = B
6	3, 5	eqtri 2644	1 |- U. ran { <. A , B >. } = B

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   U.cuni 4436   ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  elxp4  7110  elxp5  7111  op2nd  7177  fo2nd  7189  f2ndres  7191  ixpsnf1o  7948  xpassen  8054  xpdom2  8055