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Theorem 2nd1st 5932
Description: Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)
Assertion
Ref Expression
2nd1st  |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  <. ( 2nd `  A ) ,  ( 1st `  A
) >. )

Proof of Theorem 2nd1st
StepHypRef Expression
1 1st2nd2 5927 . . . . 5  |-  ( A  e.  ( B  X.  C )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
21sneqd 3454 . . . 4  |-  ( A  e.  ( B  X.  C )  ->  { A }  =  { <. ( 1st `  A ) ,  ( 2nd `  A
) >. } )
32cnveqd 4600 . . 3  |-  ( A  e.  ( B  X.  C )  ->  `' { A }  =  `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. } )
43unieqd 3659 . 2  |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  U. `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. } )
5 1stexg 5920 . . 3  |-  ( A  e.  ( B  X.  C )  ->  ( 1st `  A )  e. 
_V )
6 2ndexg 5921 . . 3  |-  ( A  e.  ( B  X.  C )  ->  ( 2nd `  A )  e. 
_V )
7 opswapg 4904 . . 3  |-  ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V )  ->  U. `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. }  =  <. ( 2nd `  A
) ,  ( 1st `  A ) >. )
85, 6, 7syl2anc 403 . 2  |-  ( A  e.  ( B  X.  C )  ->  U. `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. }  =  <. ( 2nd `  A
) ,  ( 1st `  A ) >. )
94, 8eqtrd 2120 1  |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  <. ( 2nd `  A ) ,  ( 1st `  A
) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   _Vcvv 2619   {csn 3441   <.cop 3444   U.cuni 3648    X. cxp 4426   `'ccnv 4427   ` cfv 5002   1stc1st 5891   2ndc2nd 5892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-1st 5893  df-2nd 5894
This theorem is referenced by: (None)
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