Theorem op2nda 5499

Description: Extract the second member of an ordered pair. (See op1sta 5496 to extract the first member, op2ndb 5498 for an alternate version, and op2nd 6808 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 5495	. . 3 |- ran { <. A , B >. } = { B }
3	2	unieqi 4260	. 2 |- U. ran { <. A , B >. } = U. { B }
4		cnvsn.2	. . 3 |- B e. _V
5	4	unisn 4266	. 2 |- U. { B } = B
6	3, 5	eqtri 2486	1 |- U. ran { <. A , B >. } = B

Syntax hints:   = wceq 1395   e. wcel 1819   _Vcvv 3109   {csn 4032   <.cop 4038   U.cuni 4251   ran crn 5009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-dm 5018  df-rn 5019

This theorem is referenced by:  elxp4  6743  elxp5  6744  op2nd  6808  fo2nd  6820  f2ndres  6822  ixpsnf1o  7528  xpassen  7630  xpdom2  7631  xpnnenOLD  13955