Theorem op2nda 3458

Description: Extract the second member of an ordered pair. (See op1sta 3454 to extract the first member, op2ndb 3457 for an alternate version, and op2nd 4092 for the preferred 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 3195	. . . 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 3455	. . . . 5 |- `'{<.A, B>.} = {<.B, A>.}
5	4	dmeqi 3318	. . . 4 |- dom `'{<.A, B>.} = dom {<.B, A>.}
6	1, 5	eqtr 1498	. . 3 |- ran {<.A, B>.} = dom {<.B, A>.}
7	6	unieqi 2515	. 2 |- U.ran {<.A, B>.} = U.dom {<.B, A>.}
8	3	op1sta 3454	. 2 |- U.dom {<.B, A>.} = B
9	7, 8	eqtr 1498	1 |- U.ran {<.A, B>.} = B

Syntax hints:   = wceq 958   e. wcel 960  Vcvv 1814  {csn 2413  <.cop 2415  U.cuni 2507  `'ccnv 3175  dom cdm 3176  ran crn 3177

This theorem is referenced by:  elxp4 3459  elxp5 3460  op2nd 4092  fo2nd 4098  f2ndres 4100  xpassen 4447  xpdom2 4448  xpmapenlem2 4503  xpmapenlem4 4505  xpmapenlem5 4506  mapunen 4508  xpnnen 7500

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195

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