Theorem op2ndg 6375

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 3899	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	fveq2d 5694	. . 3 |- ( x = A -> ( 2nd ` <. x , y >. ) = ( 2nd ` <. A , y >. ) )
3	2	eqeq1d 2247	. 2 |- ( x = A -> ( ( 2nd ` <. x , y >. ) = y <-> ( 2nd ` <. A , y >. ) = y ) )
4		opeq2 3900	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
5	4	fveq2d 5694	. . 3 |- ( y = B -> ( 2nd ` <. A , y >. ) = ( 2nd ` <. A , B >. ) )
6		id 19	. . 3 |- ( y = B -> y = B )
7	5, 6	eqeq12d 2253	. 2 |- ( y = B -> ( ( 2nd ` <. A , y >. ) = y <-> ( 2nd ` <. A , B >. ) = B ) )
8		vex 2824	. . 3 |- x e. _V
9		vex 2824	. . 3 |- y e. _V
10	8, 9	op2nd 6371	. 2 |- ( 2nd ` <. x , y >. ) = y
11	3, 7, 10	vtocl2g 2887	1 |- ( ( A e. V /\ B e. W ) -> ( 2nd ` <. A , B >. ) = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   <.cop 3708   ` cfv 5372   2ndc2nd 6363

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fv 5380  df-2nd 6365

This theorem is referenced by:  ot2ndg  6377  ot3rdgg  6378  2ndconst  6448  xpmapenlem  7139  2ndinl  7405  2ndinr  7407  mulpipq  7729  suplocexprlem2b  8071  aprcl  8964  frec2uzrdg  10824  frecuzrdgsuc  10829  swrdval  11398  eucalglt  12813  eucalg  12815  qredeu  12853  sqpweven  12931  2sqpwodd  12932  qnumdenbi  12948  upxp  15296  uptx  15298  txmetcnp  15542  opiedgfv  16180