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Theorem fvsng 5712
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 3778 . . . . 5  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
21sneqd 3605 . . . 4  |-  ( a  =  A  ->  { <. a ,  b >. }  =  { <. A ,  b
>. } )
3 id 19 . . . 4  |-  ( a  =  A  ->  a  =  A )
42, 3fveq12d 5522 . . 3  |-  ( a  =  A  ->  ( { <. a ,  b
>. } `  a )  =  ( { <. A ,  b >. } `  A ) )
54eqeq1d 2186 . 2  |-  ( a  =  A  ->  (
( { <. a ,  b >. } `  a )  =  b  <-> 
( { <. A , 
b >. } `  A
)  =  b ) )
6 opeq2 3779 . . . . 5  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
76sneqd 3605 . . . 4  |-  ( b  =  B  ->  { <. A ,  b >. }  =  { <. A ,  B >. } )
87fveq1d 5517 . . 3  |-  ( b  =  B  ->  ( { <. A ,  b
>. } `  A )  =  ( { <. A ,  B >. } `  A ) )
9 id 19 . . 3  |-  ( b  =  B  ->  b  =  B )
108, 9eqeq12d 2192 . 2  |-  ( b  =  B  ->  (
( { <. A , 
b >. } `  A
)  =  b  <->  ( { <. A ,  B >. } `
 A )  =  B ) )
11 vex 2740 . . 3  |-  a  e. 
_V
12 vex 2740 . . 3  |-  b  e. 
_V
1311, 12fvsn 5711 . 2  |-  ( {
<. a ,  b >. } `  a )  =  b
145, 10, 13vtocl2g 2801 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { <. A ,  B >. } `  A
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   {csn 3592   <.cop 3595   ` cfv 5216
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224
This theorem is referenced by:  fsnunfv  5717  fvpr1g  5722  fvpr2g  5723  tfr0dm  6322  fseq1p1m1  10091  1fv  10136  sumsnf  11412  prodsnf  11595  setsslid  12507  mgm1  12743  sgrp1  12770  mnd1  12801  mnd1id  12802  grp1  12930  ring1  13189
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