Theorem op2nda 2639

Description: Extract the second member of an ordered pair. (See op1sta 2635 to extract the first member and op2ndb 2638 for an alternate version.)

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		df-rn 2429	. . . 4 |- ran {<.A, B>.} = dom `'{<.A, B>.}
2		cnvsn.1	. . . . . 6 |- A e. V
3		cnvsn.2	. . . . . 6 |- B e. V
4	2, 3	cnvsn 2636	. . . . 5 |- `'{<.A, B>.} = {<.B, A>.}
5	4	dmeqi 2532	. . . 4 |- dom `'{<.A, B>.} = dom {<.B, A>.}
6	1, 5	eqtr 1119	. . 3 |- ran {<.A, B>.} = dom {<.B, A>.}
7	6	unieqi 1928	. 2 |- U.ran {<.A, B>.} = U.dom {<.B, A>.}
8	3	op1sta 2635	. 2 |- U.dom {<.B, A>.} = B
9	7, 8	eqtr 1119	1 |- U.ran {<.A, B>.} = B

Syntax hints:   = wceq 1091   e. wcel 1092  Vcvv 1348  {csn 1808  <.cop 1810  U.cuni 1919  `'ccnv 2409  dom cdm 2410  ran crn 2411

This theorem is referenced by:  elxp4 2640  elxp5 2641  op2nd 3092  fo2nd 3095  xpassen 3344  xpdom2 3345  xpmapenlem2 3392  xpmapenlem4 3394  xpmapenlem5 3395  mapunen 3397  xpnnen 4927

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429

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