ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fsng Unicode version

Theorem fsng 5586
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 3533 . . . 4  |-  ( a  =  A  ->  { a }  =  { A } )
21feq2d 5255 . . 3  |-  ( a  =  A  ->  ( F : { a } --> { b }  <->  F : { A } --> { b } ) )
3 opeq1 3700 . . . . 5  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
43sneqd 3535 . . . 4  |-  ( a  =  A  ->  { <. a ,  b >. }  =  { <. A ,  b
>. } )
54eqeq2d 2149 . . 3  |-  ( a  =  A  ->  ( F  =  { <. a ,  b >. }  <->  F  =  { <. A ,  b
>. } ) )
62, 5bibi12d 234 . 2  |-  ( a  =  A  ->  (
( F : {
a } --> { b }  <->  F  =  { <. a ,  b >. } )  <->  ( F : { A } --> { b }  <->  F  =  { <. A ,  b >. } ) ) )
7 sneq 3533 . . . 4  |-  ( b  =  B  ->  { b }  =  { B } )
8 feq3 5252 . . . 4  |-  ( { b }  =  { B }  ->  ( F : { A } --> { b }  <->  F : { A } --> { B } ) )
97, 8syl 14 . . 3  |-  ( b  =  B  ->  ( F : { A } --> { b }  <->  F : { A } --> { B } ) )
10 opeq2 3701 . . . . 5  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
1110sneqd 3535 . . . 4  |-  ( b  =  B  ->  { <. A ,  b >. }  =  { <. A ,  B >. } )
1211eqeq2d 2149 . . 3  |-  ( b  =  B  ->  ( F  =  { <. A , 
b >. }  <->  F  =  { <. A ,  B >. } ) )
139, 12bibi12d 234 . 2  |-  ( b  =  B  ->  (
( F : { A } --> { b }  <-> 
F  =  { <. A ,  b >. } )  <-> 
( F : { A } --> { B }  <->  F  =  { <. A ,  B >. } ) ) )
14 vex 2684 . . 3  |-  a  e. 
_V
15 vex 2684 . . 3  |-  b  e. 
_V
1614, 15fsn 5585 . 2  |-  ( F : { a } --> { b }  <->  F  =  { <. a ,  b
>. } )
176, 13, 16vtocl2g 2745 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( F : { A } --> { B }  <->  F  =  { <. A ,  B >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   {csn 3522   <.cop 3525   -->wf 5114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125
This theorem is referenced by:  fsn2  5587  xpsng  5588  ftpg  5597  fseq1p1m1  9867
  Copyright terms: Public domain W3C validator