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Theorem fsng 5713
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 3664 . . . 4  |-  ( a  =  A  ->  { a }  =  { A } )
21feq2d 5396 . . 3  |-  ( a  =  A  ->  ( F : { a } --> { b }  <->  F : { A } --> { b } ) )
3 opeq1 3812 . . . . 5  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
43sneqd 3666 . . . 4  |-  ( a  =  A  ->  { <. a ,  b >. }  =  { <. A ,  b
>. } )
54eqeq2d 2307 . . 3  |-  ( a  =  A  ->  ( F  =  { <. a ,  b >. }  <->  F  =  { <. A ,  b
>. } ) )
62, 5bibi12d 312 . 2  |-  ( a  =  A  ->  (
( F : {
a } --> { b }  <->  F  =  { <. a ,  b >. } )  <->  ( F : { A } --> { b }  <->  F  =  { <. A ,  b >. } ) ) )
7 sneq 3664 . . . 4  |-  ( b  =  B  ->  { b }  =  { B } )
8 feq3 5393 . . . 4  |-  ( { b }  =  { B }  ->  ( F : { A } --> { b }  <->  F : { A } --> { B } ) )
97, 8syl 15 . . 3  |-  ( b  =  B  ->  ( F : { A } --> { b }  <->  F : { A } --> { B } ) )
10 opeq2 3813 . . . . 5  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
1110sneqd 3666 . . . 4  |-  ( b  =  B  ->  { <. A ,  b >. }  =  { <. A ,  B >. } )
1211eqeq2d 2307 . . 3  |-  ( b  =  B  ->  ( F  =  { <. A , 
b >. }  <->  F  =  { <. A ,  B >. } ) )
139, 12bibi12d 312 . 2  |-  ( b  =  B  ->  (
( F : { A } --> { b }  <-> 
F  =  { <. A ,  b >. } )  <-> 
( F : { A } --> { B }  <->  F  =  { <. A ,  B >. } ) ) )
14 vex 2804 . . 3  |-  a  e. 
_V
15 vex 2804 . . 3  |-  b  e. 
_V
1614, 15fsn 5712 . 2  |-  ( F : { a } --> { b }  <->  F  =  { <. a ,  b
>. } )
176, 13, 16vtocl2g 2860 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( F : { A } --> { B }  <->  F  =  { <. A ,  B >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653   <.cop 3656   -->wf 5267
This theorem is referenced by:  xpsng  5715  axdc3lem4  8095  fseq1p1m1  10873  cats1un  11492  rngosn3  21109  esumsn  23452  bnj149  29223
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-reu 2563  df-rab 2565  df-v 2803  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-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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