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Theorem 2nd1st 6828
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 6820 . . . . 5  |-  ( A  e.  ( B  X.  C )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
21sneqd 3983 . . . 4  |-  ( A  e.  ( B  X.  C )  ->  { A }  =  { <. ( 1st `  A ) ,  ( 2nd `  A
) >. } )
32cnveqd 4998 . . 3  |-  ( A  e.  ( B  X.  C )  ->  `' { A }  =  `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. } )
43unieqd 4200 . 2  |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  U. `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. } )
5 opswap 5310 . 2  |-  U. `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. }  =  <. ( 2nd `  A
) ,  ( 1st `  A ) >.
64, 5syl6eq 2459 1  |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  <. ( 2nd `  A ) ,  ( 1st `  A
) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842   {csn 3971   <.cop 3977   U.cuni 4190    X. cxp 4820   `'ccnv 4821   ` cfv 5568   1stc1st 6781   2ndc2nd 6782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-iota 5532  df-fun 5570  df-fv 5576  df-1st 6783  df-2nd 6784
This theorem is referenced by:  fcnvgreu  27943
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