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Theorem fsng 5454
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 3452 . . . 4  |-  ( a  =  A  ->  { a }  =  { A } )
21feq2d 5136 . . 3  |-  ( a  =  A  ->  ( F : { a } --> { b }  <->  F : { A } --> { b } ) )
3 opeq1 3617 . . . . 5  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
43sneqd 3454 . . . 4  |-  ( a  =  A  ->  { <. a ,  b >. }  =  { <. A ,  b
>. } )
54eqeq2d 2099 . . 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 3452 . . . 4  |-  ( b  =  B  ->  { b }  =  { B } )
8 feq3 5133 . . . 4  |-  ( { b }  =  { B }  ->  ( F : { A } --> { b }  <->  F : { A } --> { B } ) )
97, 8syl 14 . . 3  |-  ( b  =  B  ->  ( F : { A } --> { b }  <->  F : { A } --> { B } ) )
10 opeq2 3618 . . . . 5  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
1110sneqd 3454 . . . 4  |-  ( b  =  B  ->  { <. A ,  b >. }  =  { <. A ,  B >. } )
1211eqeq2d 2099 . . 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 2622 . . 3  |-  a  e. 
_V
15 vex 2622 . . 3  |-  b  e. 
_V
1614, 15fsn 5453 . 2  |-  ( F : { a } --> { b }  <->  F  =  { <. a ,  b
>. } )
176, 13, 16vtocl2g 2683 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 1289    e. wcel 1438   {csn 3441   <.cop 3444   -->wf 4998
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-reu 2366  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  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-rn 4439  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009
This theorem is referenced by:  fsn2  5455  xpsng  5456  ftpg  5465  fseq1p1m1  9475
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