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Theorem 2nd1st 6826
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 6818 . . . . 5  |-  ( A  e.  ( B  X.  C )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
21sneqd 4039 . . . 4  |-  ( A  e.  ( B  X.  C )  ->  { A }  =  { <. ( 1st `  A ) ,  ( 2nd `  A
) >. } )
32cnveqd 5176 . . 3  |-  ( A  e.  ( B  X.  C )  ->  `' { A }  =  `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. } )
43unieqd 4255 . 2  |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  U. `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. } )
5 opswap 5493 . 2  |-  U. `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. }  =  <. ( 2nd `  A
) ,  ( 1st `  A ) >.
64, 5syl6eq 2524 1  |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  <. ( 2nd `  A ) ,  ( 1st `  A
) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {csn 4027   <.cop 4033   U.cuni 4245    X. cxp 4997   `'ccnv 4998   ` cfv 5586   1stc1st 6779   2ndc2nd 6780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fv 5594  df-1st 6781  df-2nd 6782
This theorem is referenced by:  fcnvgreu  27186
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