Theorem op2ndg 6236

Description: Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Assertion

Ref	Expression
op2ndg	|- ( ( A e. V /\ B e. W ) -> ( 2nd ` <. A , B >. ) = B )

Proof of Theorem op2ndg

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3818	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	fveq2d 5579	. . 3 |- ( x = A -> ( 2nd ` <. x , y >. ) = ( 2nd ` <. A , y >. ) )
3	2	eqeq1d 2213	. 2 |- ( x = A -> ( ( 2nd ` <. x , y >. ) = y <-> ( 2nd ` <. A , y >. ) = y ) )
4		opeq2 3819	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
5	4	fveq2d 5579	. . 3 |- ( y = B -> ( 2nd ` <. A , y >. ) = ( 2nd ` <. A , B >. ) )
6		id 19	. . 3 |- ( y = B -> y = B )
7	5, 6	eqeq12d 2219	. 2 |- ( y = B -> ( ( 2nd ` <. A , y >. ) = y <-> ( 2nd ` <. A , B >. ) = B ) )
8		vex 2774	. . 3 |- x e. _V
9		vex 2774	. . 3 |- y e. _V
10	8, 9	op2nd 6232	. 2 |- ( 2nd ` <. x , y >. ) = y
11	3, 7, 10	vtocl2g 2836	1 |- ( ( A e. V /\ B e. W ) -> ( 2nd ` <. A , B >. ) = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   <.cop 3635   ` cfv 5270   2ndc2nd 6224

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-iota 5231  df-fun 5272  df-fv 5278  df-2nd 6226

This theorem is referenced by:  ot2ndg  6238  ot3rdgg  6239  2ndconst  6307  xpmapenlem  6945  2ndinl  7176  2ndinr  7178  mulpipq  7484  suplocexprlem2b  7826  aprcl  8718  frec2uzrdg  10552  frecuzrdgsuc  10557  eucalglt  12350  eucalg  12352  qredeu  12390  sqpweven  12468  2sqpwodd  12469  qnumdenbi  12485  upxp  14715  uptx  14717  txmetcnp  14961  opiedgfv  15593