Theorem op1stbg 4600

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 3881	. . . . 5 |- ( ( A e. V /\ B e. W ) -> <. A , B >. = { { A } , { A , B } } )
2	1	inteqd 3954	. . . 4 |- ( ( A e. V /\ B e. W ) -> |^| <. A , B >. = |^| { { A } , { A , B } } )
3		snexg 4297	. . . . . 6 |- ( A e. V -> { A } e. _V )
4		prexg 4325	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )
5		intprg 3982	. . . . . 6 |- ( ( { A } e. _V /\ { A , B } e. _V ) -> |^| { { A } , { A , B } } = ( { A } i^i { A , B } ) )
6	3, 4, 5	syl2an2r 599	. . . . 5 |- ( ( A e. V /\ B e. W ) -> |^| { { A } , { A , B } } = ( { A } i^i { A , B } ) )
7		snsspr1 3842	. . . . . 6 |- { A } C_ { A , B }
8		df-ss 3224	. . . . . 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 2281	. . . 4 |- ( ( A e. V /\ B e. W ) -> |^| { { A } , { A , B } } = { A } )
11	2, 10	eqtrd 2265	. . 3 |- ( ( A e. V /\ B e. W ) -> |^| <. A , B >. = { A } )
12	11	inteqd 3954	. 2 |- ( ( A e. V /\ B e. W ) -> |^| |^| <. A , B >. = |^| { A } )
13		intsng 3983	. . 3 |- ( A e. V -> |^| { A } = A )
14	13	adantr 276	. 2 |- ( ( A e. V /\ B e. W ) -> |^| { A } = A )
15	12, 14	eqtrd 2265	1 |- ( ( A e. V /\ B e. W ) -> |^| |^| <. A , B >. = A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   _Vcvv 2813   i^i cin 3210   C_ wss 3211   {csn 3689   {cpr 3690   <.cop 3692   |^|cint 3949

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-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-int 3950

This theorem is referenced by:  elxp5  5251  fundmen  7047