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Theorem funopsn 38881
Description: If a function is an ordered pair then it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)
Hypotheses
Ref Expression
funopsn.x  |-  X  e. 
_V
funopsn.y  |-  Y  e. 
_V
Assertion
Ref Expression
funopsn  |-  ( ( Fun  F  /\  F  =  <. X ,  Y >. )  ->  E. a
( X  =  {
a }  /\  F  =  { <. a ,  a
>. } ) )
Distinct variable groups:    F, a    X, a    Y, a

Proof of Theorem funopsn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 funiun 38880 . . 3  |-  ( Fun 
F  ->  F  =  U_ x  e.  dom  F { <. x ,  ( F `  x )
>. } )
2 eqeq1 2426 . . . . . . . . . 10  |-  ( F  =  <. X ,  Y >.  ->  ( F  = 
U_ x  e.  dom  F { <. x ,  ( F `  x )
>. }  <->  <. X ,  Y >.  =  U_ x  e. 
dom  F { <. x ,  ( F `  x ) >. } ) )
3 eqcom 2431 . . . . . . . . . 10  |-  ( <. X ,  Y >.  = 
U_ x  e.  dom  F { <. x ,  ( F `  x )
>. }  <->  U_ x  e.  dom  F { <. x ,  ( F `  x )
>. }  =  <. X ,  Y >. )
42, 3syl6bb 264 . . . . . . . . 9  |-  ( F  =  <. X ,  Y >.  ->  ( F  = 
U_ x  e.  dom  F { <. x ,  ( F `  x )
>. }  <->  U_ x  e.  dom  F { <. x ,  ( F `  x )
>. }  =  <. X ,  Y >. ) )
54adantl 467 . . . . . . . 8  |-  ( ( Fun  F  /\  F  =  <. X ,  Y >. )  ->  ( F  =  U_ x  e.  dom  F { <. x ,  ( F `  x )
>. }  <->  U_ x  e.  dom  F { <. x ,  ( F `  x )
>. }  =  <. X ,  Y >. ) )
6 funopsn.x . . . . . . . . . . 11  |-  X  e. 
_V
7 funopsn.y . . . . . . . . . . 11  |-  Y  e. 
_V
86, 7opnzi 4693 . . . . . . . . . 10  |-  <. X ,  Y >.  =/=  (/)
9 neeq1 2701 . . . . . . . . . . . . . 14  |-  ( <. X ,  Y >.  =  F  ->  ( <. X ,  Y >.  =/=  (/)  <->  F  =/=  (/) ) )
109eqcoms 2434 . . . . . . . . . . . . 13  |-  ( F  =  <. X ,  Y >.  ->  ( <. X ,  Y >.  =/=  (/)  <->  F  =/=  (/) ) )
11 funrel 5618 . . . . . . . . . . . . . . . . 17  |-  ( Fun 
F  ->  Rel  F )
12 reldm0 5071 . . . . . . . . . . . . . . . . 17  |-  ( Rel 
F  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
1311, 12syl 17 . . . . . . . . . . . . . . . 16  |-  ( Fun 
F  ->  ( F  =  (/)  <->  dom  F  =  (/) ) )
1413biimprd 226 . . . . . . . . . . . . . . 15  |-  ( Fun 
F  ->  ( dom  F  =  (/)  ->  F  =  (/) ) )
1514necon3d 2644 . . . . . . . . . . . . . 14  |-  ( Fun 
F  ->  ( F  =/=  (/)  ->  dom  F  =/=  (/) ) )
1615com12 32 . . . . . . . . . . . . 13  |-  ( F  =/=  (/)  ->  ( Fun  F  ->  dom  F  =/=  (/) ) )
1710, 16syl6bi 231 . . . . . . . . . . . 12  |-  ( F  =  <. X ,  Y >.  ->  ( <. X ,  Y >.  =/=  (/)  ->  ( Fun  F  ->  dom  F  =/=  (/) ) ) )
1817com3l 84 . . . . . . . . . . 11  |-  ( <. X ,  Y >.  =/=  (/)  ->  ( Fun  F  ->  ( F  =  <. X ,  Y >.  ->  dom  F  =/=  (/) ) ) )
1918impd 432 . . . . . . . . . 10  |-  ( <. X ,  Y >.  =/=  (/)  ->  ( ( Fun 
F  /\  F  =  <. X ,  Y >. )  ->  dom  F  =/=  (/) ) )
208, 19ax-mp 5 . . . . . . . . 9  |-  ( ( Fun  F  /\  F  =  <. X ,  Y >. )  ->  dom  F  =/=  (/) )
21 fvex 5891 . . . . . . . . . 10  |-  ( F `
 x )  e. 
_V
2221, 6, 7iunopeqop 38870 . . . . . . . . 9  |-  ( dom 
F  =/=  (/)  ->  ( U_ x  e.  dom  F { <. x ,  ( F `  x )
>. }  =  <. X ,  Y >.  ->  E. a dom  F  =  { a } ) )
2320, 22syl 17 . . . . . . . 8  |-  ( ( Fun  F  /\  F  =  <. X ,  Y >. )  ->  ( U_ x  e.  dom  F { <. x ,  ( F `
 x ) >. }  =  <. X ,  Y >.  ->  E. a dom  F  =  { a } ) )
245, 23sylbid 218 . . . . . . 7  |-  ( ( Fun  F  /\  F  =  <. X ,  Y >. )  ->  ( F  =  U_ x  e.  dom  F { <. x ,  ( F `  x )
>. }  ->  E. a dom  F  =  { a } ) )
2524imp 430 . . . . . 6  |-  ( ( ( Fun  F  /\  F  =  <. X ,  Y >. )  /\  F  =  U_ x  e.  dom  F { <. x ,  ( F `  x )
>. } )  ->  E. a dom  F  =  { a } )
26 iuneq1 4313 . . . . . . . . . . . 12  |-  ( dom 
F  =  { a }  ->  U_ x  e. 
dom  F { <. x ,  ( F `  x ) >. }  =  U_ x  e.  { a }  { <. x ,  ( F `  x ) >. } )
27 vex 3083 . . . . . . . . . . . . 13  |-  a  e. 
_V
28 id 22 . . . . . . . . . . . . . . 15  |-  ( x  =  a  ->  x  =  a )
29 fveq2 5881 . . . . . . . . . . . . . . 15  |-  ( x  =  a  ->  ( F `  x )  =  ( F `  a ) )
3028, 29opeq12d 4195 . . . . . . . . . . . . . 14  |-  ( x  =  a  ->  <. x ,  ( F `  x ) >.  =  <. a ,  ( F `  a ) >. )
3130sneqd 4010 . . . . . . . . . . . . 13  |-  ( x  =  a  ->  { <. x ,  ( F `  x ) >. }  =  { <. a ,  ( F `  a )
>. } )
3227, 31iunxsn 4382 . . . . . . . . . . . 12  |-  U_ x  e.  { a }  { <. x ,  ( F `
 x ) >. }  =  { <. a ,  ( F `  a ) >. }
3326, 32syl6eq 2479 . . . . . . . . . . 11  |-  ( dom 
F  =  { a }  ->  U_ x  e. 
dom  F { <. x ,  ( F `  x ) >. }  =  { <. a ,  ( F `  a )
>. } )
3433adantl 467 . . . . . . . . . 10  |-  ( ( ( Fun  F  /\  F  =  <. X ,  Y >. )  /\  dom  F  =  { a } )  ->  U_ x  e. 
dom  F { <. x ,  ( F `  x ) >. }  =  { <. a ,  ( F `  a )
>. } )
3534eqeq2d 2436 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  F  =  <. X ,  Y >. )  /\  dom  F  =  { a } )  ->  ( F  =  U_ x  e.  dom  F { <. x ,  ( F `  x )
>. }  <->  F  =  { <. a ,  ( F `
 a ) >. } ) )
36 eqeq1 2426 . . . . . . . . . . . . . 14  |-  ( F  =  <. X ,  Y >.  ->  ( F  =  { <. a ,  ( F `  a )
>. }  <->  <. X ,  Y >.  =  { <. a ,  ( F `  a ) >. } ) )
3736adantl 467 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\  F  =  <. X ,  Y >. )  ->  ( F  =  { <. a ,  ( F `  a )
>. }  <->  <. X ,  Y >.  =  { <. a ,  ( F `  a ) >. } ) )
38 eqcom 2431 . . . . . . . . . . . . . 14  |-  ( <. X ,  Y >.  =  { <. a ,  ( F `  a )
>. }  <->  { <. a ,  ( F `  a )
>. }  =  <. X ,  Y >. )
39 fvex 5891 . . . . . . . . . . . . . . 15  |-  ( F `
 a )  e. 
_V
4027, 39, 6, 7snopeqop 38864 . . . . . . . . . . . . . 14  |-  ( {
<. a ,  ( F `
 a ) >. }  =  <. X ,  Y >. 
<->  ( a  =  ( F `  a )  /\  X  =  Y  /\  X  =  {
a } ) )
4138, 40sylbb 200 . . . . . . . . . . . . 13  |-  ( <. X ,  Y >.  =  { <. a ,  ( F `  a )
>. }  ->  ( a  =  ( F `  a )  /\  X  =  Y  /\  X  =  { a } ) )
4237, 41syl6bi 231 . . . . . . . . . . . 12  |-  ( ( Fun  F  /\  F  =  <. X ,  Y >. )  ->  ( F  =  { <. a ,  ( F `  a )
>. }  ->  ( a  =  ( F `  a )  /\  X  =  Y  /\  X  =  { a } ) ) )
4342imp 430 . . . . . . . . . . 11  |-  ( ( ( Fun  F  /\  F  =  <. X ,  Y >. )  /\  F  =  { <. a ,  ( F `  a )
>. } )  ->  (
a  =  ( F `
 a )  /\  X  =  Y  /\  X  =  { a } ) )
44 simpr3 1013 . . . . . . . . . . . . . . 15  |-  ( ( F  =  { <. a ,  ( F `  a ) >. }  /\  ( a  =  ( F `  a )  /\  X  =  Y  /\  X  =  {
a } ) )  ->  X  =  {
a } )
45 simp1 1005 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( a  =  ( F `
 a )  /\  X  =  Y  /\  X  =  { a } )  ->  a  =  ( F `  a ) )
4645eqcomd 2430 . . . . . . . . . . . . . . . . . . 19  |-  ( ( a  =  ( F `
 a )  /\  X  =  Y  /\  X  =  { a } )  ->  ( F `  a )  =  a )
4746opeq2d 4194 . . . . . . . . . . . . . . . . . 18  |-  ( ( a  =  ( F `
 a )  /\  X  =  Y  /\  X  =  { a } )  ->  <. a ,  ( F `  a ) >.  =  <. a ,  a >. )
4847sneqd 4010 . . . . . . . . . . . . . . . . 17  |-  ( ( a  =  ( F `
 a )  /\  X  =  Y  /\  X  =  { a } )  ->  { <. a ,  ( F `  a ) >. }  =  { <. a ,  a
>. } )
4948eqeq2d 2436 . . . . . . . . . . . . . . . 16  |-  ( ( a  =  ( F `
 a )  /\  X  =  Y  /\  X  =  { a } )  ->  ( F  =  { <. a ,  ( F `  a ) >. }  <->  F  =  { <. a ,  a
>. } ) )
5049biimpac 488 . . . . . . . . . . . . . . 15  |-  ( ( F  =  { <. a ,  ( F `  a ) >. }  /\  ( a  =  ( F `  a )  /\  X  =  Y  /\  X  =  {
a } ) )  ->  F  =  { <. a ,  a >. } )
5144, 50jca 534 . . . . . . . . . . . . . 14  |-  ( ( F  =  { <. a ,  ( F `  a ) >. }  /\  ( a  =  ( F `  a )  /\  X  =  Y  /\  X  =  {
a } ) )  ->  ( X  =  { a }  /\  F  =  { <. a ,  a >. } ) )
5251ex 435 . . . . . . . . . . . . 13  |-  ( F  =  { <. a ,  ( F `  a ) >. }  ->  ( ( a  =  ( F `  a )  /\  X  =  Y  /\  X  =  {
a } )  -> 
( X  =  {
a }  /\  F  =  { <. a ,  a
>. } ) ) )
5352adantl 467 . . . . . . . . . . . 12  |-  ( ( ( Fun  F  /\  F  =  <. X ,  Y >. )  /\  F  =  { <. a ,  ( F `  a )
>. } )  ->  (
( a  =  ( F `  a )  /\  X  =  Y  /\  X  =  {
a } )  -> 
( X  =  {
a }  /\  F  =  { <. a ,  a
>. } ) ) )
5453a1dd 47 . . . . . . . . . . 11  |-  ( ( ( Fun  F  /\  F  =  <. X ,  Y >. )  /\  F  =  { <. a ,  ( F `  a )
>. } )  ->  (
( a  =  ( F `  a )  /\  X  =  Y  /\  X  =  {
a } )  -> 
( dom  F  =  { a }  ->  ( X  =  { a }  /\  F  =  { <. a ,  a
>. } ) ) ) )
5543, 54mpd 15 . . . . . . . . . 10  |-  ( ( ( Fun  F  /\  F  =  <. X ,  Y >. )  /\  F  =  { <. a ,  ( F `  a )
>. } )  ->  ( dom  F  =  { a }  ->  ( X  =  { a }  /\  F  =  { <. a ,  a >. } ) ) )
5655impancom 441 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  F  =  <. X ,  Y >. )  /\  dom  F  =  { a } )  ->  ( F  =  { <. a ,  ( F `  a )
>. }  ->  ( X  =  { a }  /\  F  =  { <. a ,  a >. } ) ) )
5735, 56sylbid 218 . . . . . . . 8  |-  ( ( ( Fun  F  /\  F  =  <. X ,  Y >. )  /\  dom  F  =  { a } )  ->  ( F  =  U_ x  e.  dom  F { <. x ,  ( F `  x )
>. }  ->  ( X  =  { a }  /\  F  =  { <. a ,  a >. } ) ) )
5857impancom 441 . . . . . . 7  |-  ( ( ( Fun  F  /\  F  =  <. X ,  Y >. )  /\  F  =  U_ x  e.  dom  F { <. x ,  ( F `  x )
>. } )  ->  ( dom  F  =  { a }  ->  ( X  =  { a }  /\  F  =  { <. a ,  a >. } ) ) )
5958eximdv 1758 . . . . . 6  |-  ( ( ( Fun  F  /\  F  =  <. X ,  Y >. )  /\  F  =  U_ x  e.  dom  F { <. x ,  ( F `  x )
>. } )  ->  ( E. a dom  F  =  { a }  ->  E. a ( X  =  { a }  /\  F  =  { <. a ,  a >. } ) ) )
6025, 59mpd 15 . . . . 5  |-  ( ( ( Fun  F  /\  F  =  <. X ,  Y >. )  /\  F  =  U_ x  e.  dom  F { <. x ,  ( F `  x )
>. } )  ->  E. a
( X  =  {
a }  /\  F  =  { <. a ,  a
>. } ) )
6160expcom 436 . . . 4  |-  ( F  =  U_ x  e. 
dom  F { <. x ,  ( F `  x ) >. }  ->  ( ( Fun  F  /\  F  =  <. X ,  Y >. )  ->  E. a
( X  =  {
a }  /\  F  =  { <. a ,  a
>. } ) ) )
6261expd 437 . . 3  |-  ( F  =  U_ x  e. 
dom  F { <. x ,  ( F `  x ) >. }  ->  ( Fun  F  ->  ( F  =  <. X ,  Y >.  ->  E. a
( X  =  {
a }  /\  F  =  { <. a ,  a
>. } ) ) ) )
631, 62mpcom 37 . 2  |-  ( Fun 
F  ->  ( F  =  <. X ,  Y >.  ->  E. a ( X  =  { a }  /\  F  =  { <. a ,  a >. } ) ) )
6463imp 430 1  |-  ( ( Fun  F  /\  F  =  <. X ,  Y >. )  ->  E. a
( X  =  {
a }  /\  F  =  { <. a ,  a
>. } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872    =/= wne 2614   _Vcvv 3080   (/)c0 3761   {csn 3998   <.cop 4004   U_ciun 4299   dom cdm 4853   Rel wrel 4858   Fun wfun 5595   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609
This theorem is referenced by:  funop  38882  funop1  38883
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