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Theorem oteqimp 6757
Description: The components of an ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
Assertion
Ref Expression
oteqimp  |-  ( T  =  <. A ,  B ,  C >.  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  B  /\  ( 2nd `  T
)  =  C ) ) )

Proof of Theorem oteqimp
StepHypRef Expression
1 ot1stg 6752 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A )
2 ot2ndg 6753 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B )
3 ot3rdg 6754 . . . 4  |-  ( C  e.  Z  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
433ad2ant3 1020 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
51, 2, 43jca 1177 . 2  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A  /\  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B  /\  ( 2nd `  <. A ,  B ,  C >. )  =  C ) )
6 fveq2 5805 . . . . 5  |-  ( T  =  <. A ,  B ,  C >.  ->  ( 1st `  T )  =  ( 1st `  <. A ,  B ,  C >. ) )
76fveq2d 5809 . . . 4  |-  ( T  =  <. A ,  B ,  C >.  ->  ( 1st `  ( 1st `  T
) )  =  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) ) )
87eqeq1d 2404 . . 3  |-  ( T  =  <. A ,  B ,  C >.  ->  ( ( 1st `  ( 1st `  T ) )  =  A  <->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A ) )
96fveq2d 5809 . . . 4  |-  ( T  =  <. A ,  B ,  C >.  ->  ( 2nd `  ( 1st `  T
) )  =  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) ) )
109eqeq1d 2404 . . 3  |-  ( T  =  <. A ,  B ,  C >.  ->  ( ( 2nd `  ( 1st `  T ) )  =  B  <->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B ) )
11 fveq2 5805 . . . 4  |-  ( T  =  <. A ,  B ,  C >.  ->  ( 2nd `  T )  =  ( 2nd `  <. A ,  B ,  C >. ) )
1211eqeq1d 2404 . . 3  |-  ( T  =  <. A ,  B ,  C >.  ->  ( ( 2nd `  T )  =  C  <->  ( 2nd ` 
<. A ,  B ,  C >. )  =  C ) )
138, 10, 123anbi123d 1301 . 2  |-  ( T  =  <. A ,  B ,  C >.  ->  ( ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  B  /\  ( 2nd `  T
)  =  C )  <-> 
( ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A  /\  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B  /\  ( 2nd `  <. A ,  B ,  C >. )  =  C ) ) )
145, 13syl5ibr 221 1  |-  ( T  =  <. A ,  B ,  C >.  ->  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( ( 1st `  ( 1st `  T ) )  =  A  /\  ( 2nd `  ( 1st `  T
) )  =  B  /\  ( 2nd `  T
)  =  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1405    e. wcel 1842   <.cotp 3979   ` cfv 5525   1stc1st 6736   2ndc2nd 6737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-ot 3980  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-iota 5489  df-fun 5527  df-fv 5533  df-1st 6738  df-2nd 6739
This theorem is referenced by:  el2wlkonot  25167  el2spthonot  25168  el2spthonot0  25169
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