Theorem op2ndg 6057

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 3713	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	fveq2d 5433	. . 3 |- ( x = A -> ( 2nd ` <. x , y >. ) = ( 2nd ` <. A , y >. ) )
3	2	eqeq1d 2149	. 2 |- ( x = A -> ( ( 2nd ` <. x , y >. ) = y <-> ( 2nd ` <. A , y >. ) = y ) )
4		opeq2 3714	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
5	4	fveq2d 5433	. . 3 |- ( y = B -> ( 2nd ` <. A , y >. ) = ( 2nd ` <. A , B >. ) )
6		id 19	. . 3 |- ( y = B -> y = B )
7	5, 6	eqeq12d 2155	. 2 |- ( y = B -> ( ( 2nd ` <. A , y >. ) = y <-> ( 2nd ` <. A , B >. ) = B ) )
8		vex 2692	. . 3 |- x e. _V
9		vex 2692	. . 3 |- y e. _V
10	8, 9	op2nd 6053	. 2 |- ( 2nd ` <. x , y >. ) = y
11	3, 7, 10	vtocl2g 2753	1 |- ( ( A e. V /\ B e. W ) -> ( 2nd ` <. A , B >. ) = B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   <.cop 3535   ` cfv 5131   2ndc2nd 6045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fv 5139  df-2nd 6047

This theorem is referenced by:  ot2ndg  6059  ot3rdgg  6060  2ndconst  6127  xpmapenlem  6751  2ndinl  6968  2ndinr  6970  mulpipq  7204  suplocexprlem2b  7546  aprcl  8432  frec2uzrdg  10213  frecuzrdgsuc  10218  eucalglt  11774  eucalg  11776  qredeu  11814  sqpweven  11889  2sqpwodd  11890  qnumdenbi  11906  upxp  12480  uptx  12482  txmetcnp  12726