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Theorem ot1stg 6354
Description: Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 6354, ot2ndg 6355, ot3rdg 6356.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)
Assertion
Ref Expression
ot1stg  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A )

Proof of Theorem ot1stg
StepHypRef Expression
1 df-ot 3817 . . . . . 6  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5724 . . . . 5  |-  ( 1st `  <. A ,  B ,  C >. )  =  ( 1st `  <. <. A ,  B >. ,  C >. )
3 opex 4420 . . . . . 6  |-  <. A ,  B >.  e.  _V
4 op1stg 6352 . . . . . 6  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  X )  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
53, 4mpan 652 . . . . 5  |-  ( C  e.  X  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
62, 5syl5eq 2480 . . . 4  |-  ( C  e.  X  ->  ( 1st `  <. A ,  B ,  C >. )  =  <. A ,  B >. )
76fveq2d 5725 . . 3  |-  ( C  e.  X  ->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  ( 1st `  <. A ,  B >. ) )
8 op1stg 6352 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
97, 8sylan9eqr 2490 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  C  e.  X )  ->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A )
1093impa 1148 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2949   <.cop 3810   <.cotp 3811   ` cfv 5447   1stc1st 6340
This theorem is referenced by:  splval  11773  mamufval  27412  el2xptp0  28052  oteqimp  28054  frg2woteq  28387  mapdhval  32460  hdmap1val  32535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-ot 3817  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-iota 5411  df-fun 5449  df-fv 5455  df-1st 6342
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