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Theorem eqop 6729
Description: Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
eqop  |-  ( A  e.  ( V  X.  W )  ->  ( A  =  <. B ,  C >. 
<->  ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) ) )

Proof of Theorem eqop
StepHypRef Expression
1 1st2nd2 6726 . . 3  |-  ( A  e.  ( V  X.  W )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
21eqeq1d 2456 . 2  |-  ( A  e.  ( V  X.  W )  ->  ( A  =  <. B ,  C >. 
<-> 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >. )
)
3 fvex 5812 . . 3  |-  ( 1st `  A )  e.  _V
4 fvex 5812 . . 3  |-  ( 2nd `  A )  e.  _V
53, 4opth 4677 . 2  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >.  <->  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C ) )
62, 5syl6bb 261 1  |-  ( A  e.  ( V  X.  W )  ->  ( A  =  <. B ,  C >. 
<->  ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3994    X. cxp 4949   ` cfv 5529   1stc1st 6688   2ndc2nd 6689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fv 5537  df-1st 6690  df-2nd 6691
This theorem is referenced by:  eqop2  6730  op1steq  6731  lsmhash  16326  txhmeo  19511  ptuncnv  19515  rngosn3  24085  f1od2  26195  el2xptp0  30295  wlkcomp  30454  clwlkcomp  30594  dvhb1dimN  34988
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