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Theorem 2nd1st 6046
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 6041 . . . . 5  |-  ( A  e.  ( B  X.  C )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
21sneqd 3510 . . . 4  |-  ( A  e.  ( B  X.  C )  ->  { A }  =  { <. ( 1st `  A ) ,  ( 2nd `  A
) >. } )
32cnveqd 4685 . . 3  |-  ( A  e.  ( B  X.  C )  ->  `' { A }  =  `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. } )
43unieqd 3717 . 2  |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  U. `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. } )
5 1stexg 6033 . . 3  |-  ( A  e.  ( B  X.  C )  ->  ( 1st `  A )  e. 
_V )
6 2ndexg 6034 . . 3  |-  ( A  e.  ( B  X.  C )  ->  ( 2nd `  A )  e. 
_V )
7 opswapg 4995 . . 3  |-  ( ( ( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V )  ->  U. `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. }  =  <. ( 2nd `  A
) ,  ( 1st `  A ) >. )
85, 6, 7syl2anc 408 . 2  |-  ( A  e.  ( B  X.  C )  ->  U. `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. }  =  <. ( 2nd `  A
) ,  ( 1st `  A ) >. )
94, 8eqtrd 2150 1  |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  <. ( 2nd `  A ) ,  ( 1st `  A
) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    e. wcel 1465   _Vcvv 2660   {csn 3497   <.cop 3500   U.cuni 3706    X. cxp 4507   `'ccnv 4508   ` cfv 5093   1stc1st 6004   2ndc2nd 6005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101  df-1st 6006  df-2nd 6007
This theorem is referenced by: (None)
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