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Theorem fvsng 5859
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 3919 . . . . 5  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
21sneqd 3763 . . . 4  |-  ( a  =  A  ->  { <. a ,  b >. }  =  { <. A ,  b
>. } )
3 id 20 . . . 4  |-  ( a  =  A  ->  a  =  A )
42, 3fveq12d 5667 . . 3  |-  ( a  =  A  ->  ( { <. a ,  b
>. } `  a )  =  ( { <. A ,  b >. } `  A ) )
54eqeq1d 2388 . 2  |-  ( a  =  A  ->  (
( { <. a ,  b >. } `  a )  =  b  <-> 
( { <. A , 
b >. } `  A
)  =  b ) )
6 opeq2 3920 . . . . 5  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
76sneqd 3763 . . . 4  |-  ( b  =  B  ->  { <. A ,  b >. }  =  { <. A ,  B >. } )
87fveq1d 5663 . . 3  |-  ( b  =  B  ->  ( { <. A ,  b
>. } `  A )  =  ( { <. A ,  B >. } `  A ) )
9 id 20 . . 3  |-  ( b  =  B  ->  b  =  B )
108, 9eqeq12d 2394 . 2  |-  ( b  =  B  ->  (
( { <. A , 
b >. } `  A
)  =  b  <->  ( { <. A ,  B >. } `
 A )  =  B ) )
11 vex 2895 . . 3  |-  a  e. 
_V
12 vex 2895 . . 3  |-  b  e. 
_V
1311, 12fvsn 5858 . 2  |-  ( {
<. a ,  b >. } `  a )  =  b
145, 10, 13vtocl2g 2951 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { <. A ,  B >. } `  A
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {csn 3750   <.cop 3753   ` cfv 5387
This theorem is referenced by:  fsnunfv  5865  fvpr1g  5869  fvpr2g  5870  axdc3lem4  8259  1fv  11043  fseq1p1m1  11045  s1fv  11680  sumsn  12454  seq1st  12982  vdwlem8  13276  setsid  13428  xpsc0  13705  xpsc1  13706  gsumws1  14705  dprdsn  15514  frgpcyg  16770  pt1hmeo  17752  vdgr1d  21515  vdgr1b  21516  vdgr1a  21518  eupap1  21539  cvmliftlem7  24750  cvmliftlem13  24755  prodsn  25058  enfixsn  26919  sumsnd  27358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395
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