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Theorem fsng 5368
Description: A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.)
Assertion
Ref Expression
fsng  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( F : { A } --> { B }  <->  F  =  { <. A ,  B >. } ) )

Proof of Theorem fsng
Dummy variables  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sneq 3417 . . . 4  |-  ( a  =  A  ->  { a }  =  { A } )
21feq2d 5066 . . 3  |-  ( a  =  A  ->  ( F : { a } --> { b }  <->  F : { A } --> { b } ) )
3 opeq1 3578 . . . . 5  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
43sneqd 3419 . . . 4  |-  ( a  =  A  ->  { <. a ,  b >. }  =  { <. A ,  b
>. } )
54eqeq2d 2093 . . 3  |-  ( a  =  A  ->  ( F  =  { <. a ,  b >. }  <->  F  =  { <. A ,  b
>. } ) )
62, 5bibi12d 233 . 2  |-  ( a  =  A  ->  (
( F : {
a } --> { b }  <->  F  =  { <. a ,  b >. } )  <->  ( F : { A } --> { b }  <->  F  =  { <. A ,  b >. } ) ) )
7 sneq 3417 . . . 4  |-  ( b  =  B  ->  { b }  =  { B } )
8 feq3 5063 . . . 4  |-  ( { b }  =  { B }  ->  ( F : { A } --> { b }  <->  F : { A } --> { B } ) )
97, 8syl 14 . . 3  |-  ( b  =  B  ->  ( F : { A } --> { b }  <->  F : { A } --> { B } ) )
10 opeq2 3579 . . . . 5  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
1110sneqd 3419 . . . 4  |-  ( b  =  B  ->  { <. A ,  b >. }  =  { <. A ,  B >. } )
1211eqeq2d 2093 . . 3  |-  ( b  =  B  ->  ( F  =  { <. A , 
b >. }  <->  F  =  { <. A ,  B >. } ) )
139, 12bibi12d 233 . 2  |-  ( b  =  B  ->  (
( F : { A } --> { b }  <-> 
F  =  { <. A ,  b >. } )  <-> 
( F : { A } --> { B }  <->  F  =  { <. A ,  B >. } ) ) )
14 vex 2605 . . 3  |-  a  e. 
_V
15 vex 2605 . . 3  |-  b  e. 
_V
1614, 15fsn 5367 . 2  |-  ( F : { a } --> { b }  <->  F  =  { <. a ,  b
>. } )
176, 13, 16vtocl2g 2663 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( F : { A } --> { B }  <->  F  =  { <. A ,  B >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   {csn 3406   <.cop 3409   -->wf 4928
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939
This theorem is referenced by:  fsn2  5369  xpsng  5370  ftpg  5379  fseq1p1m1  9187
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