Theorem fvsng 5834

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 3856	. . . . 5 |- ( a = A -> <. a , b >. = <. A , b >. )
2	1	sneqd 3679	. . . 4 |- ( a = A -> { <. a , b >. } = { <. A , b >. } )
3		id 19	. . . 4 |- ( a = A -> a = A )
4	2, 3	fveq12d 5633	. . 3 |- ( a = A -> ( { <. a , b >. } ` a ) = ( { <. A , b >. } ` A ) )
5	4	eqeq1d 2238	. 2 |- ( a = A -> ( ( { <. a , b >. } ` a ) = b <-> ( { <. A , b >. } ` A ) = b ) )
6		opeq2 3857	. . . . 5 |- ( b = B -> <. A , b >. = <. A , B >. )
7	6	sneqd 3679	. . . 4 |- ( b = B -> { <. A , b >. } = { <. A , B >. } )
8	7	fveq1d 5628	. . 3 |- ( b = B -> ( { <. A , b >. } ` A ) = ( { <. A , B >. } ` A ) )
9		id 19	. . 3 |- ( b = B -> b = B )
10	8, 9	eqeq12d 2244	. 2 |- ( b = B -> ( ( { <. A , b >. } ` A ) = b <-> ( { <. A , B >. } ` A ) = B ) )
11		vex 2802	. . 3 |- a e. _V
12		vex 2802	. . 3 |- b e. _V
13	11, 12	fvsn 5833	. 2 |- ( { <. a , b >. } ` a ) = b
14	5, 10, 13	vtocl2g 2865	1 |- ( ( A e. V /\ B e. W ) -> ( { <. A , B >. } ` A ) = B )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   {csn 3666   <.cop 3669   ` cfv 5317

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325

This theorem is referenced by:  fsnunfv  5839  fvpr1g  5844  fvpr2g  5845  tfr0dm  6466  fseq1p1m1  10286  1fv  10331  s1fv  11154  sumsnf  11915  prodsnf  12098  setsslid  13078  mgm1  13398  sgrp1  13439  mnd1  13483  mnd1id  13484  grp1  13634  ring1  14017  ixpsnbasval  14424