MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  el2xptp0 Structured version   Unicode version

Theorem el2xptp0 6843
Description: A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
Assertion
Ref Expression
el2xptp0  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  ->  ( ( A  e.  ( ( U  X.  V )  X.  W
)  /\  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y  /\  ( 2nd `  A )  =  Z ) )  <->  A  =  <. X ,  Y ,  Z >. ) )

Proof of Theorem el2xptp0
StepHypRef Expression
1 xp1st 6829 . . . . . 6  |-  ( A  e.  ( ( U  X.  V )  X.  W )  ->  ( 1st `  A )  e.  ( U  X.  V
) )
21ad2antrl 727 . . . . 5  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  ( ( U  X.  V )  X.  W
)  /\  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y  /\  ( 2nd `  A )  =  Z ) ) )  ->  ( 1st `  A
)  e.  ( U  X.  V ) )
3 3simpa 993 . . . . . . 7  |-  ( ( ( 1st `  ( 1st `  A ) )  =  X  /\  ( 2nd `  ( 1st `  A
) )  =  Y  /\  ( 2nd `  A
)  =  Z )  ->  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y ) )
43adantl 466 . . . . . 6  |-  ( ( A  e.  ( ( U  X.  V )  X.  W )  /\  ( ( 1st `  ( 1st `  A ) )  =  X  /\  ( 2nd `  ( 1st `  A
) )  =  Y  /\  ( 2nd `  A
)  =  Z ) )  ->  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y ) )
54adantl 466 . . . . 5  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  ( ( U  X.  V )  X.  W
)  /\  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y  /\  ( 2nd `  A )  =  Z ) ) )  ->  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y ) )
6 eqopi 6833 . . . . 5  |-  ( ( ( 1st `  A
)  e.  ( U  X.  V )  /\  ( ( 1st `  ( 1st `  A ) )  =  X  /\  ( 2nd `  ( 1st `  A
) )  =  Y ) )  ->  ( 1st `  A )  = 
<. X ,  Y >. )
72, 5, 6syl2anc 661 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  ( ( U  X.  V )  X.  W
)  /\  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y  /\  ( 2nd `  A )  =  Z ) ) )  ->  ( 1st `  A
)  =  <. X ,  Y >. )
8 simprr3 1046 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  ( ( U  X.  V )  X.  W
)  /\  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y  /\  ( 2nd `  A )  =  Z ) ) )  ->  ( 2nd `  A
)  =  Z )
97, 8jca 532 . . 3  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  ( ( U  X.  V )  X.  W
)  /\  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y  /\  ( 2nd `  A )  =  Z ) ) )  ->  ( ( 1st `  A )  =  <. X ,  Y >.  /\  ( 2nd `  A )  =  Z ) )
10 df-ot 4041 . . . . . 6  |-  <. X ,  Y ,  Z >.  = 
<. <. X ,  Y >. ,  Z >.
1110eqeq2i 2475 . . . . 5  |-  ( A  =  <. X ,  Y ,  Z >.  <->  A  =  <. <. X ,  Y >. ,  Z >. )
12 eqop 6839 . . . . 5  |-  ( A  e.  ( ( U  X.  V )  X.  W )  ->  ( A  =  <. <. X ,  Y >. ,  Z >.  <->  (
( 1st `  A
)  =  <. X ,  Y >.  /\  ( 2nd `  A )  =  Z ) ) )
1311, 12syl5bb 257 . . . 4  |-  ( A  e.  ( ( U  X.  V )  X.  W )  ->  ( A  =  <. X ,  Y ,  Z >.  <->  (
( 1st `  A
)  =  <. X ,  Y >.  /\  ( 2nd `  A )  =  Z ) ) )
1413ad2antrl 727 . . 3  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  ( ( U  X.  V )  X.  W
)  /\  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y  /\  ( 2nd `  A )  =  Z ) ) )  ->  ( A  = 
<. X ,  Y ,  Z >. 
<->  ( ( 1st `  A
)  =  <. X ,  Y >.  /\  ( 2nd `  A )  =  Z ) ) )
159, 14mpbird 232 . 2  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  ( A  e.  ( ( U  X.  V )  X.  W
)  /\  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y  /\  ( 2nd `  A )  =  Z ) ) )  ->  A  =  <. X ,  Y ,  Z >. )
16 opelxpi 5040 . . . . . . . 8  |-  ( ( X  e.  U  /\  Y  e.  V )  -> 
<. X ,  Y >.  e.  ( U  X.  V
) )
17163adant3 1016 . . . . . . 7  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  -> 
<. X ,  Y >.  e.  ( U  X.  V
) )
18 simp3 998 . . . . . . 7  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  ->  Z  e.  W )
19 opelxp 5038 . . . . . . 7  |-  ( <. <. X ,  Y >. ,  Z >.  e.  (
( U  X.  V
)  X.  W )  <-> 
( <. X ,  Y >.  e.  ( U  X.  V )  /\  Z  e.  W ) )
2017, 18, 19sylanbrc 664 . . . . . 6  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  -> 
<. <. X ,  Y >. ,  Z >.  e.  ( ( U  X.  V
)  X.  W ) )
2110, 20syl5eqel 2549 . . . . 5  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  -> 
<. X ,  Y ,  Z >.  e.  ( ( U  X.  V )  X.  W ) )
2221adantr 465 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  A  =  <. X ,  Y ,  Z >. )  ->  <. X ,  Y ,  Z >.  e.  ( ( U  X.  V )  X.  W
) )
23 eleq1 2529 . . . . 5  |-  ( A  =  <. X ,  Y ,  Z >.  ->  ( A  e.  ( ( U  X.  V )  X.  W )  <->  <. X ,  Y ,  Z >.  e.  ( ( U  X.  V )  X.  W
) ) )
2423adantl 466 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  A  =  <. X ,  Y ,  Z >. )  ->  ( A  e.  ( ( U  X.  V )  X.  W )  <->  <. X ,  Y ,  Z >.  e.  ( ( U  X.  V )  X.  W
) ) )
2522, 24mpbird 232 . . 3  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  A  =  <. X ,  Y ,  Z >. )  ->  A  e.  ( ( U  X.  V )  X.  W
) )
26 fveq2 5872 . . . . . 6  |-  ( A  =  <. X ,  Y ,  Z >.  ->  ( 1st `  A )  =  ( 1st `  <. X ,  Y ,  Z >. ) )
2726fveq2d 5876 . . . . 5  |-  ( A  =  <. X ,  Y ,  Z >.  ->  ( 1st `  ( 1st `  A
) )  =  ( 1st `  ( 1st `  <. X ,  Y ,  Z >. ) ) )
28 ot1stg 6813 . . . . 5  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  ->  ( 1st `  ( 1st `  <. X ,  Y ,  Z >. ) )  =  X )
2927, 28sylan9eqr 2520 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  A  =  <. X ,  Y ,  Z >. )  ->  ( 1st `  ( 1st `  A
) )  =  X )
3026fveq2d 5876 . . . . 5  |-  ( A  =  <. X ,  Y ,  Z >.  ->  ( 2nd `  ( 1st `  A
) )  =  ( 2nd `  ( 1st `  <. X ,  Y ,  Z >. ) ) )
31 ot2ndg 6814 . . . . 5  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  ->  ( 2nd `  ( 1st `  <. X ,  Y ,  Z >. ) )  =  Y )
3230, 31sylan9eqr 2520 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  A  =  <. X ,  Y ,  Z >. )  ->  ( 2nd `  ( 1st `  A
) )  =  Y )
33 fveq2 5872 . . . . 5  |-  ( A  =  <. X ,  Y ,  Z >.  ->  ( 2nd `  A )  =  ( 2nd `  <. X ,  Y ,  Z >. ) )
34 ot3rdg 6815 . . . . . 6  |-  ( Z  e.  W  ->  ( 2nd `  <. X ,  Y ,  Z >. )  =  Z )
35343ad2ant3 1019 . . . . 5  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  ->  ( 2nd `  <. X ,  Y ,  Z >. )  =  Z )
3633, 35sylan9eqr 2520 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  A  =  <. X ,  Y ,  Z >. )  ->  ( 2nd `  A )  =  Z )
3729, 32, 363jca 1176 . . 3  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  A  =  <. X ,  Y ,  Z >. )  ->  (
( 1st `  ( 1st `  A ) )  =  X  /\  ( 2nd `  ( 1st `  A
) )  =  Y  /\  ( 2nd `  A
)  =  Z ) )
3825, 37jca 532 . 2  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  A  =  <. X ,  Y ,  Z >. )  ->  ( A  e.  ( ( U  X.  V )  X.  W )  /\  (
( 1st `  ( 1st `  A ) )  =  X  /\  ( 2nd `  ( 1st `  A
) )  =  Y  /\  ( 2nd `  A
)  =  Z ) ) )
3915, 38impbida 832 1  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  ->  ( ( A  e.  ( ( U  X.  V )  X.  W
)  /\  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y  /\  ( 2nd `  A )  =  Z ) )  <->  A  =  <. X ,  Y ,  Z >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   <.cop 4038   <.cotp 4040    X. cxp 5006   ` cfv 5594   1stc1st 6797   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-ot 4041  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fv 5602  df-1st 6799  df-2nd 6800
This theorem is referenced by:  el2wlkonot  24995  el2spthonot  24996
  Copyright terms: Public domain W3C validator