Theorem op2ndg 6204

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 3804	. . . 4 |- ( x = A -> <. x , y >. = <. A , y >. )
2	1	fveq2d 5558	. . 3 |- ( x = A -> ( 2nd ` <. x , y >. ) = ( 2nd ` <. A , y >. ) )
3	2	eqeq1d 2202	. 2 |- ( x = A -> ( ( 2nd ` <. x , y >. ) = y <-> ( 2nd ` <. A , y >. ) = y ) )
4		opeq2 3805	. . . 4 |- ( y = B -> <. A , y >. = <. A , B >. )
5	4	fveq2d 5558	. . 3 |- ( y = B -> ( 2nd ` <. A , y >. ) = ( 2nd ` <. A , B >. ) )
6		id 19	. . 3 |- ( y = B -> y = B )
7	5, 6	eqeq12d 2208	. 2 |- ( y = B -> ( ( 2nd ` <. A , y >. ) = y <-> ( 2nd ` <. A , B >. ) = B ) )
8		vex 2763	. . 3 |- x e. _V
9		vex 2763	. . 3 |- y e. _V
10	8, 9	op2nd 6200	. 2 |- ( 2nd ` <. x , y >. ) = y
11	3, 7, 10	vtocl2g 2824	1 |- ( ( A e. V /\ B e. W ) -> ( 2nd ` <. A , B >. ) = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   <.cop 3621   ` cfv 5254   2ndc2nd 6192

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fv 5262  df-2nd 6194

This theorem is referenced by:  ot2ndg  6206  ot3rdgg  6207  2ndconst  6275  xpmapenlem  6905  2ndinl  7134  2ndinr  7136  mulpipq  7432  suplocexprlem2b  7774  aprcl  8665  frec2uzrdg  10480  frecuzrdgsuc  10485  eucalglt  12195  eucalg  12197  qredeu  12235  sqpweven  12313  2sqpwodd  12314  qnumdenbi  12330  upxp  14440  uptx  14442  txmetcnp  14686