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Theorem oteqimp 30300
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 6704 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A )
2 ot2ndg 6705 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B )
3 ot3rdg 6706 . . . 4  |-  ( C  e.  Z  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
433ad2ant3 1011 . . 3  |-  ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  ->  ( 2nd `  <. A ,  B ,  C >. )  =  C )
51, 2, 43jca 1168 . 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 5802 . . . . 5  |-  ( T  =  <. A ,  B ,  C >.  ->  ( 1st `  T )  =  ( 1st `  <. A ,  B ,  C >. ) )
76fveq2d 5806 . . . 4  |-  ( T  =  <. A ,  B ,  C >.  ->  ( 1st `  ( 1st `  T
) )  =  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) ) )
87eqeq1d 2456 . . 3  |-  ( T  =  <. A ,  B ,  C >.  ->  ( ( 1st `  ( 1st `  T ) )  =  A  <->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A ) )
96fveq2d 5806 . . . 4  |-  ( T  =  <. A ,  B ,  C >.  ->  ( 2nd `  ( 1st `  T
) )  =  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) ) )
109eqeq1d 2456 . . 3  |-  ( T  =  <. A ,  B ,  C >.  ->  ( ( 2nd `  ( 1st `  T ) )  =  B  <->  ( 2nd `  ( 1st `  <. A ,  B ,  C >. ) )  =  B ) )
11 fveq2 5802 . . . 4  |-  ( T  =  <. A ,  B ,  C >.  ->  ( 2nd `  T )  =  ( 2nd `  <. A ,  B ,  C >. ) )
1211eqeq1d 2456 . . 3  |-  ( T  =  <. A ,  B ,  C >.  ->  ( ( 2nd `  T )  =  C  <->  ( 2nd ` 
<. A ,  B ,  C >. )  =  C ) )
138, 10, 123anbi123d 1290 . 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 965    = wceq 1370    e. wcel 1758   <.cotp 3996   ` cfv 5529   1stc1st 6688   2ndc2nd 6689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-ot 3997  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fv 5537  df-1st 6690  df-2nd 6691
This theorem is referenced by:  el2wlkonot  30559  el2spthonot  30560  el2spthonot0  30561
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