Theorem fvsng 5779

Description: The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)

Assertion

Ref	Expression
fvsng	|- ( ( A e. V /\ B e. W ) -> ( { <. A , B >. } ` A ) = B )

Proof of Theorem fvsng

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3818	. . . . 5 |- ( a = A -> <. a , b >. = <. A , b >. )
2	1	sneqd 3645	. . . 4 |- ( a = A -> { <. a , b >. } = { <. A , b >. } )
3		id 19	. . . 4 |- ( a = A -> a = A )
4	2, 3	fveq12d 5582	. . 3 |- ( a = A -> ( { <. a , b >. } ` a ) = ( { <. A , b >. } ` A ) )
5	4	eqeq1d 2213	. 2 |- ( a = A -> ( ( { <. a , b >. } ` a ) = b <-> ( { <. A , b >. } ` A ) = b ) )
6		opeq2 3819	. . . . 5 |- ( b = B -> <. A , b >. = <. A , B >. )
7	6	sneqd 3645	. . . 4 |- ( b = B -> { <. A , b >. } = { <. A , B >. } )
8	7	fveq1d 5577	. . 3 |- ( b = B -> ( { <. A , b >. } ` A ) = ( { <. A , B >. } ` A ) )
9		id 19	. . 3 |- ( b = B -> b = B )
10	8, 9	eqeq12d 2219	. 2 |- ( b = B -> ( ( { <. A , b >. } ` A ) = b <-> ( { <. A , B >. } ` A ) = B ) )
11		vex 2774	. . 3 |- a e. _V
12		vex 2774	. . 3 |- b e. _V
13	11, 12	fvsn 5778	. 2 |- ( { <. a , b >. } ` a ) = b
14	5, 10, 13	vtocl2g 2836	1 |- ( ( A e. V /\ B e. W ) -> ( { <. A , B >. } ` A ) = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   {csn 3632   <.cop 3635   ` cfv 5270

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-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

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-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278

This theorem is referenced by:  fsnunfv  5784  fvpr1g  5789  fvpr2g  5790  tfr0dm  6407  fseq1p1m1  10215  1fv  10260  s1fv  11078  sumsnf  11691  prodsnf  11874  setsslid  12854  mgm1  13173  sgrp1  13214  mnd1  13258  mnd1id  13259  grp1  13409  ring1  13792  ixpsnbasval  14199