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Theorem opiota 6810
Description: The property of a uniquely specified ordered pair. The proof uses properties of the  iota description binder. (Contributed by Mario Carneiro, 21-May-2015.)
Hypotheses
Ref Expression
opiota.1  |-  I  =  ( iota z E. x  e.  A  E. y  e.  B  (
z  =  <. x ,  y >.  /\  ph ) )
opiota.2  |-  X  =  ( 1st `  I
)
opiota.3  |-  Y  =  ( 2nd `  I
)
opiota.4  |-  ( x  =  C  ->  ( ph 
<->  ps ) )
opiota.5  |-  ( y  =  D  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
opiota  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  (
( C  e.  A  /\  D  e.  B  /\  ch )  <->  ( C  =  X  /\  D  =  Y ) ) )
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    ch, y    ph, z    x, D, y, z    ps, x
Allowed substitution hints:    ph( x, y)    ps( y, z)    ch( x, z)    I( x, y, z)    X( x, y, z)    Y( x, y, z)

Proof of Theorem opiota
StepHypRef Expression
1 opiota.4 . . . . . . 7  |-  ( x  =  C  ->  ( ph 
<->  ps ) )
2 opiota.5 . . . . . . 7  |-  ( y  =  D  ->  ( ps 
<->  ch ) )
31, 2ceqsrex2v 3149 . . . . . 6  |-  ( ( C  e.  A  /\  D  e.  B )  ->  ( E. x  e.  A  E. y  e.  B  ( ( x  =  C  /\  y  =  D )  /\  ph ) 
<->  ch ) )
43bicomd 204 . . . . 5  |-  ( ( C  e.  A  /\  D  e.  B )  ->  ( ch  <->  E. x  e.  A  E. y  e.  B  ( (
x  =  C  /\  y  =  D )  /\  ph ) ) )
5 opex 4628 . . . . . . . 8  |-  <. C ,  D >.  e.  _V
65a1i 11 . . . . . . 7  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  <. C ,  D >.  e.  _V )
7 id 22 . . . . . . 7  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  E! z E. x  e.  A  E. y  e.  B  ( z  =  <. x ,  y >.  /\  ph ) )
8 eqeq1 2432 . . . . . . . . . . 11  |-  ( z  =  <. C ,  D >.  ->  ( z  = 
<. x ,  y >.  <->  <. C ,  D >.  = 
<. x ,  y >.
) )
9 eqcom 2435 . . . . . . . . . . . 12  |-  ( <. C ,  D >.  = 
<. x ,  y >.  <->  <.
x ,  y >.  =  <. C ,  D >. )
10 vex 3025 . . . . . . . . . . . . 13  |-  x  e. 
_V
11 vex 3025 . . . . . . . . . . . . 13  |-  y  e. 
_V
1210, 11opth 4638 . . . . . . . . . . . 12  |-  ( <.
x ,  y >.  =  <. C ,  D >.  <-> 
( x  =  C  /\  y  =  D ) )
139, 12bitri 252 . . . . . . . . . . 11  |-  ( <. C ,  D >.  = 
<. x ,  y >.  <->  ( x  =  C  /\  y  =  D )
)
148, 13syl6bb 264 . . . . . . . . . 10  |-  ( z  =  <. C ,  D >.  ->  ( z  = 
<. x ,  y >.  <->  ( x  =  C  /\  y  =  D )
) )
1514anbi1d 709 . . . . . . . . 9  |-  ( z  =  <. C ,  D >.  ->  ( ( z  =  <. x ,  y
>.  /\  ph )  <->  ( (
x  =  C  /\  y  =  D )  /\  ph ) ) )
16152rexbidv 2885 . . . . . . . 8  |-  ( z  =  <. C ,  D >.  ->  ( E. x  e.  A  E. y  e.  B  ( z  =  <. x ,  y
>.  /\  ph )  <->  E. x  e.  A  E. y  e.  B  ( (
x  =  C  /\  y  =  D )  /\  ph ) ) )
1716adantl 467 . . . . . . 7  |-  ( ( E! z E. x  e.  A  E. y  e.  B  ( z  =  <. x ,  y
>.  /\  ph )  /\  z  =  <. C ,  D >. )  ->  ( E. x  e.  A  E. y  e.  B  ( z  =  <. x ,  y >.  /\  ph ) 
<->  E. x  e.  A  E. y  e.  B  ( ( x  =  C  /\  y  =  D )  /\  ph ) ) )
18 nfeu1 2286 . . . . . . 7  |-  F/ z E! z E. x  e.  A  E. y  e.  B  ( z  =  <. x ,  y
>.  /\  ph )
19 nfvd 1756 . . . . . . 7  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  F/ z E. x  e.  A  E. y  e.  B  ( ( x  =  C  /\  y  =  D )  /\  ph ) )
20 nfcvd 2570 . . . . . . 7  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  F/_ z <. C ,  D >. )
216, 7, 17, 18, 19, 20iota2df 5532 . . . . . 6  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  ( E. x  e.  A  E. y  e.  B  ( ( x  =  C  /\  y  =  D )  /\  ph ) 
<->  ( iota z E. x  e.  A  E. y  e.  B  (
z  =  <. x ,  y >.  /\  ph ) )  =  <. C ,  D >. )
)
22 eqcom 2435 . . . . . . 7  |-  ( <. C ,  D >.  =  I  <->  I  =  <. C ,  D >. )
23 opiota.1 . . . . . . . 8  |-  I  =  ( iota z E. x  e.  A  E. y  e.  B  (
z  =  <. x ,  y >.  /\  ph ) )
2423eqeq1i 2433 . . . . . . 7  |-  ( I  =  <. C ,  D >.  <-> 
( iota z E. x  e.  A  E. y  e.  B  ( z  =  <. x ,  y
>.  /\  ph ) )  =  <. C ,  D >. )
2522, 24bitri 252 . . . . . 6  |-  ( <. C ,  D >.  =  I  <->  ( iota z E. x  e.  A  E. y  e.  B  ( z  =  <. x ,  y >.  /\  ph ) )  =  <. C ,  D >. )
2621, 25syl6bbr 266 . . . . 5  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  ( E. x  e.  A  E. y  e.  B  ( ( x  =  C  /\  y  =  D )  /\  ph ) 
<-> 
<. C ,  D >.  =  I ) )
274, 26sylan9bbr 705 . . . 4  |-  ( ( E! z E. x  e.  A  E. y  e.  B  ( z  =  <. x ,  y
>.  /\  ph )  /\  ( C  e.  A  /\  D  e.  B
) )  ->  ( ch 
<-> 
<. C ,  D >.  =  I ) )
2827pm5.32da 645 . . 3  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  (
( ( C  e.  A  /\  D  e.  B )  /\  ch ) 
<->  ( ( C  e.  A  /\  D  e.  B )  /\  <. C ,  D >.  =  I ) ) )
29 opelxpi 4828 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  ( A  X.  B
) )
30 simpl 458 . . . . . . . . . . 11  |-  ( ( z  =  <. x ,  y >.  /\  ph )  ->  z  =  <. x ,  y >. )
3130eleq1d 2490 . . . . . . . . . 10  |-  ( ( z  =  <. x ,  y >.  /\  ph )  ->  ( z  e.  ( A  X.  B
)  <->  <. x ,  y
>.  e.  ( A  X.  B ) ) )
3229, 31syl5ibrcom 225 . . . . . . . . 9  |-  ( ( x  e.  A  /\  y  e.  B )  ->  ( ( z  = 
<. x ,  y >.  /\  ph )  ->  z  e.  ( A  X.  B
) ) )
3332rexlimivv 2861 . . . . . . . 8  |-  ( E. x  e.  A  E. y  e.  B  (
z  =  <. x ,  y >.  /\  ph )  ->  z  e.  ( A  X.  B ) )
3433abssi 3479 . . . . . . 7  |-  { z  |  E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph ) }  C_  ( A  X.  B
)
35 iotacl 5531 . . . . . . 7  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  ( iota z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph ) )  e. 
{ z  |  E. x  e.  A  E. y  e.  B  (
z  =  <. x ,  y >.  /\  ph ) } )
3634, 35sseldi 3405 . . . . . 6  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  ( iota z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph ) )  e.  ( A  X.  B
) )
3723, 36syl5eqel 2510 . . . . 5  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  I  e.  ( A  X.  B
) )
38 opelxp 4826 . . . . . 6  |-  ( <. C ,  D >.  e.  ( A  X.  B
)  <->  ( C  e.  A  /\  D  e.  B ) )
39 eleq1 2494 . . . . . 6  |-  ( <. C ,  D >.  =  I  ->  ( <. C ,  D >.  e.  ( A  X.  B )  <-> 
I  e.  ( A  X.  B ) ) )
4038, 39syl5bbr 262 . . . . 5  |-  ( <. C ,  D >.  =  I  ->  ( ( C  e.  A  /\  D  e.  B )  <->  I  e.  ( A  X.  B ) ) )
4137, 40syl5ibrcom 225 . . . 4  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  ( <. C ,  D >.  =  I  ->  ( C  e.  A  /\  D  e.  B ) ) )
4241pm4.71rd 639 . . 3  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  ( <. C ,  D >.  =  I  <->  ( ( C  e.  A  /\  D  e.  B )  /\  <. C ,  D >.  =  I ) ) )
43 1st2nd2 6788 . . . . 5  |-  ( I  e.  ( A  X.  B )  ->  I  =  <. ( 1st `  I
) ,  ( 2nd `  I ) >. )
4437, 43syl 17 . . . 4  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  I  =  <. ( 1st `  I
) ,  ( 2nd `  I ) >. )
4544eqeq2d 2438 . . 3  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  ( <. C ,  D >.  =  I  <->  <. C ,  D >.  =  <. ( 1st `  I
) ,  ( 2nd `  I ) >. )
)
4628, 42, 453bitr2d 284 . 2  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  (
( ( C  e.  A  /\  D  e.  B )  /\  ch ) 
<-> 
<. C ,  D >.  = 
<. ( 1st `  I
) ,  ( 2nd `  I ) >. )
)
47 df-3an 984 . 2  |-  ( ( C  e.  A  /\  D  e.  B  /\  ch )  <->  ( ( C  e.  A  /\  D  e.  B )  /\  ch ) )
48 opiota.2 . . . . 5  |-  X  =  ( 1st `  I
)
4948eqeq2i 2440 . . . 4  |-  ( C  =  X  <->  C  =  ( 1st `  I ) )
50 opiota.3 . . . . 5  |-  Y  =  ( 2nd `  I
)
5150eqeq2i 2440 . . . 4  |-  ( D  =  Y  <->  D  =  ( 2nd `  I ) )
5249, 51anbi12i 701 . . 3  |-  ( ( C  =  X  /\  D  =  Y )  <->  ( C  =  ( 1st `  I )  /\  D  =  ( 2nd `  I
) ) )
53 fvex 5835 . . . 4  |-  ( 1st `  I )  e.  _V
54 fvex 5835 . . . 4  |-  ( 2nd `  I )  e.  _V
5553, 54opth2 4642 . . 3  |-  ( <. C ,  D >.  = 
<. ( 1st `  I
) ,  ( 2nd `  I ) >.  <->  ( C  =  ( 1st `  I
)  /\  D  =  ( 2nd `  I ) ) )
5652, 55bitr4i 255 . 2  |-  ( ( C  =  X  /\  D  =  Y )  <->  <. C ,  D >.  = 
<. ( 1st `  I
) ,  ( 2nd `  I ) >. )
5746, 47, 563bitr4g 291 1  |-  ( E! z E. x  e.  A  E. y  e.  B  ( z  = 
<. x ,  y >.  /\  ph )  ->  (
( C  e.  A  /\  D  e.  B  /\  ch )  <->  ( C  =  X  /\  D  =  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   E!weu 2276   {cab 2414   E.wrex 2715   _Vcvv 3022   <.cop 3947    X. cxp 4794   iotacio 5506   ` cfv 5544   1stc1st 6749   2ndc2nd 6750
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 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
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 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-iota 5508  df-fun 5546  df-fv 5552  df-1st 6751  df-2nd 6752
This theorem is referenced by:  oeeui  7258
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