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Theorem el2xptp0 30124
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 6604 . . . . . 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 id 22 . . . . . . . 8  |-  ( ( ( 1st `  ( 1st `  A ) )  =  X  /\  ( 2nd `  ( 1st `  A
) )  =  Y )  ->  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y ) )
433adant3 1008 . . . . . . 7  |-  ( ( ( 1st `  ( 1st `  A ) )  =  X  /\  ( 2nd `  ( 1st `  A
) )  =  Y  /\  ( 2nd `  A
)  =  Z )  ->  ( ( 1st `  ( 1st `  A
) )  =  X  /\  ( 2nd `  ( 1st `  A ) )  =  Y ) )
54adantl 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 ) )
65adantl 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 ) )
7 eqopi 6608 . . . . 5  |-  ( ( ( 1st `  A
)  e.  ( U  X.  V )  /\  ( ( 1st `  ( 1st `  A ) )  =  X  /\  ( 2nd `  ( 1st `  A
) )  =  Y ) )  ->  ( 1st `  A )  = 
<. X ,  Y >. )
82, 6, 7syl2anc 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 >. )
9 simprr3 1038 . . . 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 )
108, 9jca 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 ) )
11 df-ot 3884 . . . . . 6  |-  <. X ,  Y ,  Z >.  = 
<. <. X ,  Y >. ,  Z >.
1211eqeq2i 2451 . . . . 5  |-  ( A  =  <. X ,  Y ,  Z >.  <->  A  =  <. <. X ,  Y >. ,  Z >. )
13 eqop 6614 . . . . 5  |-  ( A  e.  ( ( U  X.  V )  X.  W )  ->  ( A  =  <. <. X ,  Y >. ,  Z >.  <->  (
( 1st `  A
)  =  <. X ,  Y >.  /\  ( 2nd `  A )  =  Z ) ) )
1412, 13syl5bb 257 . . . 4  |-  ( A  e.  ( ( U  X.  V )  X.  W )  ->  ( A  =  <. X ,  Y ,  Z >.  <->  (
( 1st `  A
)  =  <. X ,  Y >.  /\  ( 2nd `  A )  =  Z ) ) )
1514ad2antrl 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 ) ) )
1610, 15mpbird 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 >. )
17 opelxpi 4869 . . . . . . . 8  |-  ( ( X  e.  U  /\  Y  e.  V )  -> 
<. X ,  Y >.  e.  ( U  X.  V
) )
18173adant3 1008 . . . . . . 7  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  -> 
<. X ,  Y >.  e.  ( U  X.  V
) )
19 id 22 . . . . . . . 8  |-  ( Z  e.  W  ->  Z  e.  W )
20193ad2ant3 1011 . . . . . . 7  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  ->  Z  e.  W )
21 opelxp 4867 . . . . . . 7  |-  ( <. <. X ,  Y >. ,  Z >.  e.  (
( U  X.  V
)  X.  W )  <-> 
( <. X ,  Y >.  e.  ( U  X.  V )  /\  Z  e.  W ) )
2218, 20, 21sylanbrc 664 . . . . . 6  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  -> 
<. <. X ,  Y >. ,  Z >.  e.  ( ( U  X.  V
)  X.  W ) )
2311, 22syl5eqel 2525 . . . . 5  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  -> 
<. X ,  Y ,  Z >.  e.  ( ( U  X.  V )  X.  W ) )
2423adantr 465 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  A  =  <. X ,  Y ,  Z >. )  ->  <. X ,  Y ,  Z >.  e.  ( ( U  X.  V )  X.  W
) )
25 eleq1 2501 . . . . 5  |-  ( A  =  <. X ,  Y ,  Z >.  ->  ( A  e.  ( ( U  X.  V )  X.  W )  <->  <. X ,  Y ,  Z >.  e.  ( ( U  X.  V )  X.  W
) ) )
2625adantl 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
) ) )
2724, 26mpbird 232 . . 3  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  A  =  <. X ,  Y ,  Z >. )  ->  A  e.  ( ( U  X.  V )  X.  W
) )
28 fveq2 5689 . . . . . 6  |-  ( A  =  <. X ,  Y ,  Z >.  ->  ( 1st `  A )  =  ( 1st `  <. X ,  Y ,  Z >. ) )
2928fveq2d 5693 . . . . 5  |-  ( A  =  <. X ,  Y ,  Z >.  ->  ( 1st `  ( 1st `  A
) )  =  ( 1st `  ( 1st `  <. X ,  Y ,  Z >. ) ) )
30 ot1stg 6589 . . . . 5  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  ->  ( 1st `  ( 1st `  <. X ,  Y ,  Z >. ) )  =  X )
3129, 30sylan9eqr 2495 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  A  =  <. X ,  Y ,  Z >. )  ->  ( 1st `  ( 1st `  A
) )  =  X )
3228fveq2d 5693 . . . . 5  |-  ( A  =  <. X ,  Y ,  Z >.  ->  ( 2nd `  ( 1st `  A
) )  =  ( 2nd `  ( 1st `  <. X ,  Y ,  Z >. ) ) )
33 ot2ndg 6590 . . . . 5  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  ->  ( 2nd `  ( 1st `  <. X ,  Y ,  Z >. ) )  =  Y )
3432, 33sylan9eqr 2495 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  A  =  <. X ,  Y ,  Z >. )  ->  ( 2nd `  ( 1st `  A
) )  =  Y )
35 fveq2 5689 . . . . 5  |-  ( A  =  <. X ,  Y ,  Z >.  ->  ( 2nd `  A )  =  ( 2nd `  <. X ,  Y ,  Z >. ) )
36 ot3rdg 6591 . . . . . 6  |-  ( Z  e.  W  ->  ( 2nd `  <. X ,  Y ,  Z >. )  =  Z )
37363ad2ant3 1011 . . . . 5  |-  ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W )  ->  ( 2nd `  <. X ,  Y ,  Z >. )  =  Z )
3835, 37sylan9eqr 2495 . . . 4  |-  ( ( ( X  e.  U  /\  Y  e.  V  /\  Z  e.  W
)  /\  A  =  <. X ,  Y ,  Z >. )  ->  ( 2nd `  A )  =  Z )
3931, 34, 383jca 1168 . . 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 ) )
4027, 39jca 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 ) ) )
4116, 40impbida 828 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 965    = wceq 1369    e. wcel 1756   <.cop 3881   <.cotp 3883    X. cxp 4836   ` cfv 5416   1stc1st 6573   2ndc2nd 6574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-ot 3884  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-iota 5379  df-fun 5418  df-fv 5424  df-1st 6575  df-2nd 6576
This theorem is referenced by:  el2wlkonot  30385  el2spthonot  30386
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