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Theorem eqop2 6848
Description: Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.)
Hypotheses
Ref Expression
eqop2.1  |-  B  e. 
_V
eqop2.2  |-  C  e. 
_V
Assertion
Ref Expression
eqop2  |-  ( A  =  <. B ,  C >.  <-> 
( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) ) )

Proof of Theorem eqop2
StepHypRef Expression
1 eqop2.1 . . . 4  |-  B  e. 
_V
2 eqop2.2 . . . 4  |-  C  e. 
_V
31, 2opelvv 4900 . . 3  |-  <. B ,  C >.  e.  ( _V 
X.  _V )
4 eleq1 2495 . . 3  |-  ( A  =  <. B ,  C >.  ->  ( A  e.  ( _V  X.  _V ) 
<-> 
<. B ,  C >.  e.  ( _V  X.  _V ) ) )
53, 4mpbiri 236 . 2  |-  ( A  =  <. B ,  C >.  ->  A  e.  ( _V  X.  _V )
)
6 eqop 6847 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( A  =  <. B ,  C >.  <-> 
( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) ) )
75, 6biadan2 646 1  |-  ( A  =  <. B ,  C >.  <-> 
( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3080   <.cop 4004    X. cxp 4851   ` cfv 5601   1stc1st 6805   2ndc2nd 6806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-1st 6807  df-2nd 6808
This theorem is referenced by:  evlslem4  18730
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