Theorem ot1stg 6314

Description: Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 6314, ot2ndg 6315, ot3rdgg 6316.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Assertion

Ref	Expression
ot1stg	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` ( 1st ` <. A , B , C >. ) ) = A )

Proof of Theorem ot1stg

Step	Hyp	Ref	Expression
1		df-ot 3679	. . . . 5 |- <. A , B , C >. = <. <. A , B >. , C >.
2	1	fveq2i 5642	. . . 4 |- ( 1st ` <. A , B , C >. ) = ( 1st ` <. <. A , B >. , C >. )
3		opexg 4320	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V )
4		op1stg 6312	. . . . . 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 1220	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` <. <. A , B >. , C >. ) = <. A , B >. )
7	2, 6	eqtrid 2276	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` <. A , B , C >. ) = <. A , B >. )
8	7	fveq2d 5643	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` ( 1st ` <. A , B , C >. ) ) = ( 1st ` <. A , B >. ) )
9		op1stg 6312	. . 3 |- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A )
10	9	3adant3 1043	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` <. A , B >. ) = A )
11	8, 10	eqtrd 2264	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` ( 1st ` <. A , B , C >. ) ) = A )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   _Vcvv 2802   <.cop 3672   <.cotp 3673   ` cfv 5326   1stc1st 6300

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-1st 6302

This theorem is referenced by: (None)