Theorem ot1stg 6324

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 6324, ot2ndg 6325, ot3rdgg 6326.) (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 3683	. . . . 5 |- <. A , B , C >. = <. <. A , B >. , C >.
2	1	fveq2i 5651	. . . 4 |- ( 1st ` <. A , B , C >. ) = ( 1st ` <. <. A , B >. , C >. )
3		opexg 4326	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V )
4		op1stg 6322	. . . . . 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 1221	. . . 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 5652	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` ( 1st ` <. A , B , C >. ) ) = ( 1st ` <. A , B >. ) )
9		op1stg 6322	. . 3 |- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A )
10	9	3adant3 1044	. 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 1005   = wceq 1398   e. wcel 2202   _Vcvv 2803   <.cop 3676   <.cotp 3677   ` cfv 5333   1stc1st 6310

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-ot 3683  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fv 5341  df-1st 6312

This theorem is referenced by: (None)