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Theorem fvsng 5477
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 3617 . . . . 5  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
21sneqd 3454 . . . 4  |-  ( a  =  A  ->  { <. a ,  b >. }  =  { <. A ,  b
>. } )
3 id 19 . . . 4  |-  ( a  =  A  ->  a  =  A )
42, 3fveq12d 5296 . . 3  |-  ( a  =  A  ->  ( { <. a ,  b
>. } `  a )  =  ( { <. A ,  b >. } `  A ) )
54eqeq1d 2096 . 2  |-  ( a  =  A  ->  (
( { <. a ,  b >. } `  a )  =  b  <-> 
( { <. A , 
b >. } `  A
)  =  b ) )
6 opeq2 3618 . . . . 5  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
76sneqd 3454 . . . 4  |-  ( b  =  B  ->  { <. A ,  b >. }  =  { <. A ,  B >. } )
87fveq1d 5291 . . 3  |-  ( b  =  B  ->  ( { <. A ,  b
>. } `  A )  =  ( { <. A ,  B >. } `  A ) )
9 id 19 . . 3  |-  ( b  =  B  ->  b  =  B )
108, 9eqeq12d 2102 . 2  |-  ( b  =  B  ->  (
( { <. A , 
b >. } `  A
)  =  b  <->  ( { <. A ,  B >. } `
 A )  =  B ) )
11 vex 2622 . . 3  |-  a  e. 
_V
12 vex 2622 . . 3  |-  b  e. 
_V
1311, 12fvsn 5476 . 2  |-  ( {
<. a ,  b >. } `  a )  =  b
145, 10, 13vtocl2g 2683 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { <. A ,  B >. } `  A
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   {csn 3441   <.cop 3444   ` cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010
This theorem is referenced by:  fsnunfv  5481  fvpr1g  5485  fvpr2g  5486  tfr0dm  6069  fseq1p1m1  9475  1fv  9515  sumsnf  10766
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