Theorem ot3rdgg 6179

Description: Extract the third member of an ordered triple. (See ot1stg 6177 comment.) (Contributed by NM, 3-Apr-2015.)

Assertion

Ref	Expression
ot3rdgg	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` <. A , B , C >. ) = C )

Proof of Theorem ot3rdgg

Step	Hyp	Ref	Expression
1		df-ot 3617	. . 3 |- <. A , B , C >. = <. <. A , B >. , C >.
2	1	fveq2i 5537	. 2 |- ( 2nd ` <. A , B , C >. ) = ( 2nd ` <. <. A , B >. , C >. )
3		opexg 4246	. . . 4 |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V )
4		op2ndg 6176	. . . 4 |- ( ( <. A , B >. e. _V /\ C e. X ) -> ( 2nd ` <. <. A , B >. , C >. ) = C )
5	3, 4	sylan 283	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ C e. X ) -> ( 2nd ` <. <. A , B >. , C >. ) = C )
6	5	3impa 1196	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` <. <. A , B >. , C >. ) = C )
7	2, 6	eqtrid 2234	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` <. A , B , C >. ) = C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160   _Vcvv 2752   <.cop 3610   <.cotp 3611   ` cfv 5235   2ndc2nd 6164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-ot 3617  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fv 5243  df-2nd 6166

This theorem is referenced by: (None)