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Theorem 1st2ndb 6814
Description: Reconstruction of an ordered pair in terms of its components. (Contributed by NM, 25-Feb-2014.)
Assertion
Ref Expression
1st2ndb  |-  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )

Proof of Theorem 1st2ndb
StepHypRef Expression
1 1st2nd2 6813 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 id 22 . . 3  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
3 fvex 5869 . . . 4  |-  ( 1st `  A )  e.  _V
4 fvex 5869 . . . 4  |-  ( 2nd `  A )  e.  _V
53, 4opelvv 5040 . . 3  |-  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  ( _V 
X.  _V )
62, 5syl6eqel 2558 . 2  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  ->  A  e.  ( _V  X.  _V ) )
71, 6impbii 188 1  |-  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1374    e. wcel 1762   _Vcvv 3108   <.cop 4028    X. cxp 4992   ` cfv 5581   1stc1st 6774   2ndc2nd 6775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fv 5589  df-1st 6776  df-2nd 6777
This theorem is referenced by:  2wlkeq  24371  opfv  27146
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