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Theorem fvsng 5730
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 3812 . . . . 5  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
21sneqd 3666 . . . 4  |-  ( a  =  A  ->  { <. a ,  b >. }  =  { <. A ,  b
>. } )
3 id 19 . . . 4  |-  ( a  =  A  ->  a  =  A )
42, 3fveq12d 5547 . . 3  |-  ( a  =  A  ->  ( { <. a ,  b
>. } `  a )  =  ( { <. A ,  b >. } `  A ) )
54eqeq1d 2304 . 2  |-  ( a  =  A  ->  (
( { <. a ,  b >. } `  a )  =  b  <-> 
( { <. A , 
b >. } `  A
)  =  b ) )
6 opeq2 3813 . . . . 5  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
76sneqd 3666 . . . 4  |-  ( b  =  B  ->  { <. A ,  b >. }  =  { <. A ,  B >. } )
87fveq1d 5543 . . 3  |-  ( b  =  B  ->  ( { <. A ,  b
>. } `  A )  =  ( { <. A ,  B >. } `  A ) )
9 id 19 . . 3  |-  ( b  =  B  ->  b  =  B )
108, 9eqeq12d 2310 . 2  |-  ( b  =  B  ->  (
( { <. A , 
b >. } `  A
)  =  b  <->  ( { <. A ,  B >. } `
 A )  =  B ) )
11 vex 2804 . . 3  |-  a  e. 
_V
12 vex 2804 . . 3  |-  b  e. 
_V
1311, 12fvsn 5729 . 2  |-  ( {
<. a ,  b >. } `  a )  =  b
145, 10, 13vtocl2g 2860 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { <. A ,  B >. } `  A
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653   <.cop 3656   ` cfv 5271
This theorem is referenced by:  fsnunfv  5736  axdc3lem4  8095  fseq1p1m1  10873  s1fv  11462  sumsn  12229  seq1st  12757  vdwlem8  13051  setsid  13203  xpsc0  13478  xpsc1  13479  gsumws1  14478  dprdsn  15287  frgpcyg  16543  pt1hmeo  17513  cvmliftlem7  23837  cvmliftlem13  23842  vdgr1d  23909  vdgr1b  23910  vdgr1a  23912  eupap1  23915  cbicp  25269  enfixsn  27360  sumsnd  27800  constr3lem4  28393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279
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