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Theorem eqop 6178
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 6175 . . 3  |-  ( A  e.  ( V  X.  W )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
21eqeq1d 2304 . 2  |-  ( A  e.  ( V  X.  W )  ->  ( A  =  <. B ,  C >. 
<-> 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >. )
)
3 fvex 5555 . . 3  |-  ( 1st `  A )  e.  _V
4 fvex 5555 . . 3  |-  ( 2nd `  A )  e.  _V
53, 4opth 4261 . 2  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  =  <. B ,  C >.  <->  ( ( 1st `  A )  =  B  /\  ( 2nd `  A )  =  C ) )
62, 5syl6bb 252 1  |-  ( A  e.  ( V  X.  W )  ->  ( A  =  <. B ,  C >. 
<->  ( ( 1st `  A
)  =  B  /\  ( 2nd `  A )  =  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656    X. cxp 4703   ` cfv 5271   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  eqop2  6179  op1steq  6180  lsmhash  15030  txhmeo  17510  ptuncnv  17514  rngosn3  21109  dvhb1dimN  31797
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-1st 6138  df-2nd 6139
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