Theorem op1stbg 4570

Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.)

Assertion

Ref	Expression
op1stbg	|- ( ( A e. V /\ B e. W ) -> |^| |^| <. A , B >. = A )

Proof of Theorem op1stbg

Step	Hyp	Ref	Expression
1		dfopg 3855	. . . . 5 |- ( ( A e. V /\ B e. W ) -> <. A , B >. = { { A } , { A , B } } )
2	1	inteqd 3928	. . . 4 |- ( ( A e. V /\ B e. W ) -> |^| <. A , B >. = |^| { { A } , { A , B } } )
3		snexg 4268	. . . . . 6 |- ( A e. V -> { A } e. _V )
4		prexg 4295	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )
5		intprg 3956	. . . . . 6 |- ( ( { A } e. _V /\ { A , B } e. _V ) -> |^| { { A } , { A , B } } = ( { A } i^i { A , B } ) )
6	3, 4, 5	syl2an2r 597	. . . . 5 |- ( ( A e. V /\ B e. W ) -> |^| { { A } , { A , B } } = ( { A } i^i { A , B } ) )
7		snsspr1 3816	. . . . . 6 |- { A } C_ { A , B }
8		df-ss 3210	. . . . . 6 |- ( { A } C_ { A , B } <-> ( { A } i^i { A , B } ) = { A } )
9	7, 8	mpbi 145	. . . . 5 |- ( { A } i^i { A , B } ) = { A }
10	6, 9	eqtrdi 2278	. . . 4 |- ( ( A e. V /\ B e. W ) -> |^| { { A } , { A , B } } = { A } )
11	2, 10	eqtrd 2262	. . 3 |- ( ( A e. V /\ B e. W ) -> |^| <. A , B >. = { A } )
12	11	inteqd 3928	. 2 |- ( ( A e. V /\ B e. W ) -> |^| |^| <. A , B >. = |^| { A } )
13		intsng 3957	. . 3 |- ( A e. V -> |^| { A } = A )
14	13	adantr 276	. 2 |- ( ( A e. V /\ B e. W ) -> |^| { A } = A )
15	12, 14	eqtrd 2262	1 |- ( ( A e. V /\ B e. W ) -> |^| |^| <. A , B >. = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   i^i cin 3196   C_ wss 3197   {csn 3666   {cpr 3667   <.cop 3669   |^|cint 3923

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-int 3924

This theorem is referenced by:  elxp5  5217  fundmen  6959