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Theorem fvsng 5346
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.
StepHypRef Expression
1 opeq1 3546 . . . . 5  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
21sneqd 3385 . . . 4  |-  ( a  =  A  ->  { <. a ,  b >. }  =  { <. A ,  b
>. } )
3 id 19 . . . 4  |-  ( a  =  A  ->  a  =  A )
42, 3fveq12d 5171 . . 3  |-  ( a  =  A  ->  ( { <. a ,  b
>. } `  a )  =  ( { <. A ,  b >. } `  A ) )
54eqeq1d 2048 . 2  |-  ( a  =  A  ->  (
( { <. a ,  b >. } `  a )  =  b  <-> 
( { <. A , 
b >. } `  A
)  =  b ) )
6 opeq2 3547 . . . . 5  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
76sneqd 3385 . . . 4  |-  ( b  =  B  ->  { <. A ,  b >. }  =  { <. A ,  B >. } )
87fveq1d 5167 . . 3  |-  ( b  =  B  ->  ( { <. A ,  b
>. } `  A )  =  ( { <. A ,  B >. } `  A ) )
9 id 19 . . 3  |-  ( b  =  B  ->  b  =  B )
108, 9eqeq12d 2054 . 2  |-  ( b  =  B  ->  (
( { <. A , 
b >. } `  A
)  =  b  <->  ( { <. A ,  B >. } `
 A )  =  B ) )
11 vex 2557 . . 3  |-  a  e. 
_V
12 vex 2557 . . 3  |-  b  e. 
_V
1311, 12fvsn 5345 . 2  |-  ( {
<. a ,  b >. } `  a )  =  b
145, 10, 13vtocl2g 2614 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { <. A ,  B >. } `  A
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   {csn 3372   <.cop 3375   ` cfv 4889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-br 3762  df-opab 3816  df-id 4027  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-iota 4854  df-fun 4891  df-fv 4897
This theorem is referenced by:  fsnunfv  5350  fvpr1g  5354  fvpr2g  5355  tfr0  5924  fseq1p1m1  8923  1fv  8963
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