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Theorem fsng 5877
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 3882 . . . 4  |-  ( a  =  A  ->  { a }  =  { A } )
21feq2d 5542 . . 3  |-  ( a  =  A  ->  ( F : { a } --> { b }  <->  F : { A } --> { b } ) )
3 opeq1 4054 . . . . 5  |-  ( a  =  A  ->  <. a ,  b >.  =  <. A ,  b >. )
43sneqd 3884 . . . 4  |-  ( a  =  A  ->  { <. a ,  b >. }  =  { <. A ,  b
>. } )
54eqeq2d 2449 . . 3  |-  ( a  =  A  ->  ( F  =  { <. a ,  b >. }  <->  F  =  { <. A ,  b
>. } ) )
62, 5bibi12d 321 . 2  |-  ( a  =  A  ->  (
( F : {
a } --> { b }  <->  F  =  { <. a ,  b >. } )  <->  ( F : { A } --> { b }  <->  F  =  { <. A ,  b >. } ) ) )
7 sneq 3882 . . . 4  |-  ( b  =  B  ->  { b }  =  { B } )
8 feq3 5539 . . . 4  |-  ( { b }  =  { B }  ->  ( F : { A } --> { b }  <->  F : { A } --> { B } ) )
97, 8syl 16 . . 3  |-  ( b  =  B  ->  ( F : { A } --> { b }  <->  F : { A } --> { B } ) )
10 opeq2 4055 . . . . 5  |-  ( b  =  B  ->  <. A , 
b >.  =  <. A ,  B >. )
1110sneqd 3884 . . . 4  |-  ( b  =  B  ->  { <. A ,  b >. }  =  { <. A ,  B >. } )
1211eqeq2d 2449 . . 3  |-  ( b  =  B  ->  ( F  =  { <. A , 
b >. }  <->  F  =  { <. A ,  B >. } ) )
139, 12bibi12d 321 . 2  |-  ( b  =  B  ->  (
( F : { A } --> { b }  <-> 
F  =  { <. A ,  b >. } )  <-> 
( F : { A } --> { B }  <->  F  =  { <. A ,  B >. } ) ) )
14 vex 2970 . . 3  |-  a  e. 
_V
15 vex 2970 . . 3  |-  b  e. 
_V
1614, 15fsn 5876 . 2  |-  ( F : { a } --> { b }  <->  F  =  { <. a ,  b
>. } )
176, 13, 16vtocl2g 3029 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( F : { A } --> { B }  <->  F  =  { <. A ,  B >. } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3872   <.cop 3878   -->wf 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420
This theorem is referenced by:  xpsng  5879  ftpg  5887  axdc3lem4  8614  fseq1p1m1  11526  cats1un  12362  symg1bas  15892  rngosn3  23864  esumsn  26467  mapsnop  30687  snlindsntorlem  30893  lmod1zr  30924  bnj149  31755
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