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Theorem 2nd1st 6165
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 6159 . . . . 5  |-  ( A  e.  ( B  X.  C )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
21sneqd 3653 . . . 4  |-  ( A  e.  ( B  X.  C )  ->  { A }  =  { <. ( 1st `  A ) ,  ( 2nd `  A
) >. } )
32cnveqd 4857 . . 3  |-  ( A  e.  ( B  X.  C )  ->  `' { A }  =  `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. } )
43unieqd 3838 . 2  |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  U. `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. } )
5 opswap 5159 . 2  |-  U. `' { <. ( 1st `  A
) ,  ( 2nd `  A ) >. }  =  <. ( 2nd `  A
) ,  ( 1st `  A ) >.
64, 5syl6eq 2331 1  |-  ( A  e.  ( B  X.  C )  ->  U. `' { A }  =  <. ( 2nd `  A ) ,  ( 1st `  A
) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {csn 3640   <.cop 3643   U.cuni 3827    X. cxp 4687   `'ccnv 4688   ` cfv 5255   1stc1st 6120   2ndc2nd 6121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-1st 6122  df-2nd 6123
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