Theorem ot2ndg 6208

Description: Extract the second member of an ordered triple. (See ot1stg 6207 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 3629	. . . . 5 |- <. A , B , C >. = <. <. A , B >. , C >.
2	1	fveq2i 5558	. . . 4 |- ( 1st ` <. A , B , C >. ) = ( 1st ` <. <. A , B >. , C >. )
3		opexg 4258	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V )
4		op1stg 6205	. . . . . 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 2238	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` <. A , B , C >. ) = <. A , B >. )
8	7	fveq2d 5559	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` ( 1st ` <. A , B , C >. ) ) = ( 2nd ` <. A , B >. ) )
9		op2ndg 6206	. . 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 2226	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 2164   _Vcvv 2760   <.cop 3622   <.cotp 3623   ` cfv 5255   1stc1st 6193   2ndc2nd 6194

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-ot 3629  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fv 5263  df-1st 6195  df-2nd 6196

This theorem is referenced by: (None)