Theorem ot3rdgg 5781

Description: Extract the third member of an ordered triple. (See ot1stg 5779 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 3385	. . 3 |- <. A , B , C >. = <. <. A , B >. , C >.
2	1	fveq2i 5181	. 2 |- ( 2nd ` <. A , B , C >. ) = ( 2nd ` <. <. A , B >. , C >. )
3		opexg 3964	. . . 4 |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V )
4		op2ndg 5778	. . . 4 |- ( ( <. A , B >. e. _V /\ C e. X ) -> ( 2nd ` <. <. A , B >. , C >. ) = C )
5	3, 4	sylan 267	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ C e. X ) -> ( 2nd ` <. <. A , B >. , C >. ) = C )
6	5	3impa 1099	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` <. <. A , B >. , C >. ) = C )
7	2, 6	syl5eq 2084	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` <. A , B , C >. ) = C )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   = wceq 1243   e. wcel 1393   _Vcvv 2557   <.cop 3378   <.cotp 3379   ` cfv 4902   2ndc2nd 5766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-ot 3385  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-2nd 5768

This theorem is referenced by: (None)