Theorem ot2ndg 6211

Description: Extract the second member of an ordered triple. (See ot1stg 6210 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 3632	. . . . 5 |- <. A , B , C >. = <. <. A , B >. , C >.
2	1	fveq2i 5561	. . . 4 |- ( 1st ` <. A , B , C >. ) = ( 1st ` <. <. A , B >. , C >. )
3		opexg 4261	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V )
4		op1stg 6208	. . . . . 6 |- ( ( <. A , B >. e. _V /\ C e. X ) -> ( 1st ` <. <. A , B >. , C >. ) = <. A , B >. )
5	3, 4	sylan 283	. . . . 5 |- ( ( ( A e. V /\ B e. W ) /\ C e. X ) -> ( 1st ` <. <. A , B >. , C >. ) = <. A , B >. )
6	5	3impa 1196	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` <. <. A , B >. , C >. ) = <. A , B >. )
7	2, 6	eqtrid 2241	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` <. A , B , C >. ) = <. A , B >. )
8	7	fveq2d 5562	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` ( 1st ` <. A , B , C >. ) ) = ( 2nd ` <. A , B >. ) )
9		op2ndg 6209	. . 3 |- ( ( A e. V /\ B e. W ) -> ( 2nd ` <. A , B >. ) = B )
10	9	3adant3 1019	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` <. A , B >. ) = B )
11	8, 10	eqtrd 2229	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` ( 1st ` <. A , B , C >. ) ) = B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2167   _Vcvv 2763   <.cop 3625   <.cotp 3626   ` cfv 5258   1stc1st 6196   2ndc2nd 6197

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-ot 3632  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fv 5266  df-1st 6198  df-2nd 6199

This theorem is referenced by: (None)