Theorem op2ndb 3457

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 2919 to extract the first member, op2nda 3458 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
op2ndb	|- |^||^||^|`'{<.A, B>.} = B

Proof of Theorem op2ndb

Step	Hyp	Ref	Expression
1		cnvsn.1	. . . . . . 7 |- A e. V
2		cnvsn.2	. . . . . . 7 |- B e. V
3	1, 2	cnvsn 3455	. . . . . 6 |- `'{<.A, B>.} = {<.B, A>.}
4	3	inteqi 2541	. . . . 5 |- |^|`'{<.A, B>.} = |^|{<.B, A>.}
5		opex 2788	. . . . . 6 |- <.B, A>. e. V
6	5	intsn 2568	. . . . 5 |- |^|{<.B, A>.} = <.B, A>.
7	4, 6	eqtr 1498	. . . 4 |- |^|`'{<.A, B>.} = <.B, A>.
8	7	inteqi 2541	. . 3 |- |^||^|`'{<.A, B>.} = |^|<.B, A>.
9	8	inteqi 2541	. 2 |- |^||^||^|`'{<.A, B>.} = |^||^|<.B, A>.
10	2	op1stb 2919	. 2 |- |^||^|<.B, A>. = B
11	9, 10	eqtr 1498	1 |- |^||^||^|`'{<.A, B>.} = B

Syntax hints:   = wceq 958   e. wcel 960  Vcvv 1814  {csn 2413  <.cop 2415  |^|cint 2537  `'ccnv 3175

This theorem is referenced by:  2ndval2 4096

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-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-ral 1652  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-int 2538  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192

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