Theorem ot1stg 6205

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 6205, ot2ndg 6206, ot3rdgg 6207.) (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 3628	. . . . 5 |- <. A , B , C >. = <. <. A , B >. , C >.
2	1	fveq2i 5557	. . . 4 |- ( 1st ` <. A , B , C >. ) = ( 1st ` <. <. A , B >. , C >. )
3		opexg 4257	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. _V )
4		op1stg 6203	. . . . . 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 5558	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` ( 1st ` <. A , B , C >. ) ) = ( 1st ` <. A , B >. ) )
9		op1stg 6203	. . 3 |- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A )
10	9	3adant3 1019	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` <. A , B >. ) = A )
11	8, 10	eqtrd 2226	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( 1st ` ( 1st ` <. A , B , C >. ) ) = A )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   _Vcvv 2760   <.cop 3621   <.cotp 3622   ` cfv 5254   1stc1st 6191

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 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

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 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-ot 3628  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fv 5262  df-1st 6193

This theorem is referenced by: (None)