Theorem op2nda 5324

Description: Extract the second member of an ordered pair. (See op1sta 5321 to extract the first member, op2ndb 5323 for an alternate version, and op2nd 6807 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 5320	. . 3 |- ran { <. A , B >. } = { B }
3	2	unieqi 4210	. 2 |- U. ran { <. A , B >. } = U. { B }
4		cnvsn.2	. . 3 |- B e. _V
5	4	unisn 4216	. 2 |- U. { B } = B
6	3, 5	eqtri 2475	1 |- U. ran { <. A , B >. } = B

Syntax hints:   = wceq 1446   e. wcel 1889   _Vcvv 3047   {csn 3970   <.cop 3976   U.cuni 4201   ran crn 4838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848

This theorem is referenced by:  elxp4  6742  elxp5  6743  op2nd  6807  fo2nd  6819  f2ndres  6821  ixpsnf1o  7567  xpassen  7671  xpdom2  7672