Theorem ot3rdgg 6316

Description: Extract the third member of an ordered triple. (See ot1stg 6314 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 3679	. . 3 |- <. A , B , C >. = <. <. A , B >. , C >.
2	1	fveq2i 5642	. 2 |- ( 2nd ` <. A , B , C >. ) = ( 2nd ` <. <. A , B >. , C >. )
3		opexg 4320	. . . 4 |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V )
4		op2ndg 6313	. . . 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 1220	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` <. <. A , B >. , C >. ) = C )
7	2, 6	eqtrid 2276	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 2nd ` <. A , B , C >. ) = C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   <.cop 3672   <.cotp 3673   ` cfv 5326   2ndc2nd 6301

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-ot 3679  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-2nd 6303

This theorem is referenced by: (None)