Theorem op2ndg 6237

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 3819	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	fveq2d 5580	. . 3 |- ( x = A -> ( 2nd ` <. x , y >. ) = ( 2nd ` <. A , y >. ) )
3	2	eqeq1d 2214	. 2 |- ( x = A -> ( ( 2nd ` <. x , y >. ) = y <-> ( 2nd ` <. A , y >. ) = y ) )
4		opeq2 3820	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
5	4	fveq2d 5580	. . 3 |- ( y = B -> ( 2nd ` <. A , y >. ) = ( 2nd ` <. A , B >. ) )
6		id 19	. . 3 |- ( y = B -> y = B )
7	5, 6	eqeq12d 2220	. 2 |- ( y = B -> ( ( 2nd ` <. A , y >. ) = y <-> ( 2nd ` <. A , B >. ) = B ) )
8		vex 2775	. . 3 |- x e. _V
9		vex 2775	. . 3 |- y e. _V
10	8, 9	op2nd 6233	. 2 |- ( 2nd ` <. x , y >. ) = y
11	3, 7, 10	vtocl2g 2837	1 |- ( ( A e. V /\ B e. W ) -> ( 2nd ` <. A , B >. ) = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   <.cop 3636   ` cfv 5271   2ndc2nd 6225

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fv 5279  df-2nd 6227

This theorem is referenced by:  ot2ndg  6239  ot3rdgg  6240  2ndconst  6308  xpmapenlem  6946  2ndinl  7177  2ndinr  7179  mulpipq  7485  suplocexprlem2b  7827  aprcl  8719  frec2uzrdg  10554  frecuzrdgsuc  10559  swrdval  11101  eucalglt  12379  eucalg  12381  qredeu  12419  sqpweven  12497  2sqpwodd  12498  qnumdenbi  12514  upxp  14744  uptx  14746  txmetcnp  14990  opiedgfv  15622