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Theorem ot1stg 6016
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 6016, ot2ndg 6017, ot3rdgg 6018.) (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 3505 . . . . 5  |-  <. A ,  B ,  C >.  = 
<. <. A ,  B >. ,  C >.
21fveq2i 5390 . . . 4  |-  ( 1st `  <. A ,  B ,  C >. )  =  ( 1st `  <. <. A ,  B >. ,  C >. )
3 opexg 4118 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  e. 
_V )
4 op1stg 6014 . . . . . 6  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  X )  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
53, 4sylan 279 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  C  e.  X )  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
653impa 1159 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 1st `  <. <. A ,  B >. ,  C >. )  =  <. A ,  B >. )
72, 6syl5eq 2160 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 1st `  <. A ,  B ,  C >. )  =  <. A ,  B >. )
87fveq2d 5391 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 1st `  ( 1st `  <. A ,  B ,  C >. ) )  =  ( 1st `  <. A ,  B >. )
)
9 op1stg 6014 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1st `  <. A ,  B >. )  =  A )
1093adant3 984 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( 1st `  <. A ,  B >. )  =  A )
118, 10eqtrd 2148 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 103    /\ w3a 945    = wceq 1314    e. wcel 1463   _Vcvv 2658   <.cop 3498   <.cotp 3499   ` cfv 5091   1stc1st 6002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-ot 3505  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fv 5099  df-1st 6004
This theorem is referenced by: (None)
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