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Theorem ot1stg 6150
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 6150, ot2ndg 6151, ot3rdg 6152.) (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 3663 . . . . . 6  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5544 . . . . 5  |-  ( 1st `  <. A ,  B ,  C >. )  =  ( 1st `  <. <. A ,  B >. ,  C >. )
3 opex 4253 . . . . . 6  |-  <. A ,  B >.  e.  _V
4 op1stg 6148 . . . . . 6  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  X )  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
53, 4mpan 651 . . . . 5  |-  ( C  e.  X  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
62, 5syl5eq 2340 . . . 4  |-  ( C  e.  X  ->  ( 1st `  <. A ,  B ,  C >. )  =  <. A ,  B >. )
76fveq2d 5545 . . 3  |-  ( C  e.  X  ->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  ( 1st `  <. A ,  B >. ) )
8 op1stg 6148 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
97, 8sylan9eqr 2350 . 2  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  C  e.  X )  ->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  A )
1093impa 1146 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   <.cotp 3657   ` cfv 5271   1stc1st 6136
This theorem is referenced by:  splval  11482  mamufval  27546  mapdhval  32536  hdmap1val  32611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-ot 3663  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-1st 6138
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