Theorem ot2ndg 6044

Description: Extract the second member of an ordered triple. (See ot1stg 6043 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Assertion

Ref	Expression
ot2ndg	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` ( 1st ` <. A , B , C >. ) ) = B )

Proof of Theorem ot2ndg

Step	Hyp	Ref	Expression
1		df-ot 3532	. . . . 5 |- <. A , B , C >. = <. <. A , B >. , C >.
2	1	fveq2i 5417	. . . 4 |- ( 1st ` <. A , B , C >. ) = ( 1st ` <. <. A , B >. , C >. )
3		opexg 4145	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V )
4		op1stg 6041	. . . . . 6 |- ( ( <. A , B >. e. _V /\ C e. X ) -> ( 1st ` <. <. A , B >. , C >. ) = <. A , B >. )
5	3, 4	sylan 281	. . . . 5 |- ( ( ( A e. V /\ B e. W ) /\ C e. X ) -> ( 1st ` <. <. A , B >. , C >. ) = <. A , B >. )
6	5	3impa 1176	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` <. <. A , B >. , C >. ) = <. A , B >. )
7	2, 6	syl5eq 2182	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` <. A , B , C >. ) = <. A , B >. )
8	7	fveq2d 5418	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` ( 1st ` <. A , B , C >. ) ) = ( 2nd ` <. A , B >. ) )
9		op2ndg 6042	. . 3 |- ( ( A e. V /\ B e. W ) -> ( 2nd ` <. A , B >. ) = B )
10	9	3adant3 1001	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` <. A , B >. ) = B )
11	8, 10	eqtrd 2170	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` ( 1st ` <. A , B , C >. ) ) = B )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   _Vcvv 2681   <.cop 3525   <.cotp 3526   ` cfv 5118   1stc1st 6029   2ndc2nd 6030

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-ot 3532  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fv 5126  df-1st 6031  df-2nd 6032

This theorem is referenced by: (None)