Theorem op2nda 5484

Description: Extract the second member of an ordered pair. (See op1sta 5481 to extract the first member, op2ndb 5483 for an alternate version, and op2nd 6783 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 5480	. . 3 |- ran { <. A , B >. } = { B }
3	2	unieqi 4247	. 2 |- U. ran { <. A , B >. } = U. { B }
4		cnvsn.2	. . 3 |- B e. _V
5	4	unisn 4253	. 2 |- U. { B } = B
6	3, 5	eqtri 2489	1 |- U. ran { <. A , B >. } = B

Syntax hints:   = wceq 1374   e. wcel 1762   _Vcvv 3106   {csn 4020   <.cop 4026   U.cuni 4238   ran crn 4993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-dm 5002  df-rn 5003

This theorem is referenced by:  elxp4  6718  elxp5  6719  op2nd  6783  fo2nd  6795  f2ndres  6797  ixpsnf1o  7499  xpassen  7601  xpdom2  7602  xpnnenOLD  13793