Theorem op2ndb 3451

Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 2913 to extract the first member, op2nda 3452 for an alternate version, and op2nd 4086 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 3449	. . . . . 6 |- `'{<.A, B>.} = {<.B, A>.}
4	3	inteqi 2537	. . . . 5 |- |^|`'{<.A, B>.} = |^|{<.B, A>.}
5		opex 2782	. . . . . 6 |- <.B, A>. e. V
6	5	intsn 2564	. . . . 5 |- |^|{<.B, A>.} = <.B, A>.
7	4, 6	eqtr 1495	. . . 4 |- |^|`'{<.A, B>.} = <.B, A>.
8	7	inteqi 2537	. . 3 |- |^||^|`'{<.A, B>.} = |^|<.B, A>.
9	8	inteqi 2537	. 2 |- |^||^||^|`'{<.A, B>.} = |^||^|<.B, A>.
10	2	op1stb 2913	. 2 |- |^||^|<.B, A>. = B
11	9, 10	eqtr 1495	1 |- |^||^||^|`'{<.A, B>.} = B

Syntax hints:   = wceq 956   e. wcel 958  Vcvv 1811  {csn 2409  <.cop 2411  |^|cint 2533  `'ccnv 3169

This theorem is referenced by:  2ndval2 4090

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-int 2534  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186

Copyright terms: Public domain