MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqop Structured version   Unicode version

Theorem eqop 6813
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 6810 . . 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 5858 . . 3  |-  ( 1st `  A )  e.  _V
4 fvex 5858 . . 3  |-  ( 2nd `  A )  e.  _V
53, 4opth 4711 . 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 367    = wceq 1398    e. wcel 1823   <.cop 4022    X. cxp 4986   ` cfv 5570   1stc1st 6771   2ndc2nd 6772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-1st 6773  df-2nd 6774
This theorem is referenced by:  eqop2  6814  op1steq  6815  el2xptp0  6817  lsmhash  16922  txhmeo  20470  ptuncnv  20474  wlkcomp  24727  clwlkcomp  24965  rngosn3  25626  f1od2  27778  esum2dlem  28321  dvhb1dimN  37109
  Copyright terms: Public domain W3C validator