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Theorem funopg 4964
Description: A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
funopg  |-  ( ( A  e.  V  /\  B  e.  W  /\  Fun  <. A ,  B >. )  ->  A  =  B )

Proof of Theorem funopg
Dummy variables  u  t  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3578 . . . . 5  |-  ( u  =  A  ->  <. u ,  t >.  =  <. A ,  t >. )
21funeqd 4953 . . . 4  |-  ( u  =  A  ->  ( Fun  <. u ,  t
>. 
<->  Fun  <. A ,  t
>. ) )
3 eqeq1 2088 . . . 4  |-  ( u  =  A  ->  (
u  =  t  <->  A  =  t ) )
42, 3imbi12d 232 . . 3  |-  ( u  =  A  ->  (
( Fun  <. u ,  t >.  ->  u  =  t )  <->  ( Fun  <. A ,  t >.  ->  A  =  t )
) )
5 opeq2 3579 . . . . 5  |-  ( t  =  B  ->  <. A , 
t >.  =  <. A ,  B >. )
65funeqd 4953 . . . 4  |-  ( t  =  B  ->  ( Fun  <. A ,  t
>. 
<->  Fun  <. A ,  B >. ) )
7 eqeq2 2091 . . . 4  |-  ( t  =  B  ->  ( A  =  t  <->  A  =  B ) )
86, 7imbi12d 232 . . 3  |-  ( t  =  B  ->  (
( Fun  <. A , 
t >.  ->  A  =  t )  <->  ( Fun  <. A ,  B >.  ->  A  =  B )
) )
9 funrel 4949 . . . . 5  |-  ( Fun 
<. u ,  t >.  ->  Rel  <. u ,  t
>. )
10 vex 2605 . . . . . 6  |-  u  e. 
_V
11 vex 2605 . . . . . 6  |-  t  e. 
_V
1210, 11relop 4514 . . . . 5  |-  ( Rel 
<. u ,  t >.  <->  E. x E. y ( u  =  { x }  /\  t  =  {
x ,  y } ) )
139, 12sylib 120 . . . 4  |-  ( Fun 
<. u ,  t >.  ->  E. x E. y
( u  =  {
x }  /\  t  =  { x ,  y } ) )
1410, 11opth 4000 . . . . . . . 8  |-  ( <.
u ,  t >.  =  <. { x } ,  { x ,  y } >.  <->  ( u  =  { x }  /\  t  =  { x ,  y } ) )
15 vex 2605 . . . . . . . . . . . 12  |-  x  e. 
_V
1615opid 3596 . . . . . . . . . . 11  |-  <. x ,  x >.  =  { { x } }
1716preq1i 3480 . . . . . . . . . 10  |-  { <. x ,  x >. ,  { { x } ,  { x ,  y } } }  =  { { { x } } ,  { { x } ,  { x ,  y } } }
18 vex 2605 . . . . . . . . . . . 12  |-  y  e. 
_V
1915, 18dfop 3577 . . . . . . . . . . 11  |-  <. x ,  y >.  =  { { x } ,  { x ,  y } }
2019preq2i 3481 . . . . . . . . . 10  |-  { <. x ,  x >. ,  <. x ,  y >. }  =  { <. x ,  x >. ,  { { x } ,  { x ,  y } } }
2115snex 3965 . . . . . . . . . . 11  |-  { x }  e.  _V
22 zfpair2 3973 . . . . . . . . . . 11  |-  { x ,  y }  e.  _V
2321, 22dfop 3577 . . . . . . . . . 10  |-  <. { x } ,  { x ,  y } >.  =  { { { x } } ,  { {
x } ,  {
x ,  y } } }
2417, 20, 233eqtr4ri 2113 . . . . . . . . 9  |-  <. { x } ,  { x ,  y } >.  =  { <. x ,  x >. ,  <. x ,  y
>. }
2524eqeq2i 2092 . . . . . . . 8  |-  ( <.
u ,  t >.  =  <. { x } ,  { x ,  y } >.  <->  <. u ,  t
>.  =  { <. x ,  x >. ,  <. x ,  y >. } )
2614, 25bitr3i 184 . . . . . . 7  |-  ( ( u  =  { x }  /\  t  =  {
x ,  y } )  <->  <. u ,  t
>.  =  { <. x ,  x >. ,  <. x ,  y >. } )
27 dffun4 4943 . . . . . . . . 9  |-  ( Fun 
<. u ,  t >.  <->  ( Rel  <. u ,  t
>.  /\  A. z A. w A. v ( (
<. z ,  w >.  e. 
<. u ,  t >.  /\  <. z ,  v
>.  e.  <. u ,  t
>. )  ->  w  =  v ) ) )
2827simprbi 269 . . . . . . . 8  |-  ( Fun 
<. u ,  t >.  ->  A. z A. w A. v ( ( <.
z ,  w >.  e. 
<. u ,  t >.  /\  <. z ,  v
>.  e.  <. u ,  t
>. )  ->  w  =  v ) )
2915, 15opex 3992 . . . . . . . . . . 11  |-  <. x ,  x >.  e.  _V
3029prid1 3506 . . . . . . . . . 10  |-  <. x ,  x >.  e.  { <. x ,  x >. ,  <. x ,  y >. }
31 eleq2 2143 . . . . . . . . . 10  |-  ( <.
u ,  t >.  =  { <. x ,  x >. ,  <. x ,  y
>. }  ->  ( <. x ,  x >.  e.  <. u ,  t >.  <->  <. x ,  x >.  e.  { <. x ,  x >. ,  <. x ,  y >. } ) )
3230, 31mpbiri 166 . . . . . . . . 9  |-  ( <.
u ,  t >.  =  { <. x ,  x >. ,  <. x ,  y
>. }  ->  <. x ,  x >.  e.  <. u ,  t >. )
3315, 18opex 3992 . . . . . . . . . . 11  |-  <. x ,  y >.  e.  _V
3433prid2 3507 . . . . . . . . . 10  |-  <. x ,  y >.  e.  { <. x ,  x >. , 
<. x ,  y >. }
35 eleq2 2143 . . . . . . . . . 10  |-  ( <.
u ,  t >.  =  { <. x ,  x >. ,  <. x ,  y
>. }  ->  ( <. x ,  y >.  e.  <. u ,  t >.  <->  <. x ,  y >.  e.  { <. x ,  x >. ,  <. x ,  y >. } ) )
3634, 35mpbiri 166 . . . . . . . . 9  |-  ( <.
u ,  t >.  =  { <. x ,  x >. ,  <. x ,  y
>. }  ->  <. x ,  y >.  e.  <. u ,  t >. )
3732, 36jca 300 . . . . . . . 8  |-  ( <.
u ,  t >.  =  { <. x ,  x >. ,  <. x ,  y
>. }  ->  ( <. x ,  x >.  e.  <. u ,  t >.  /\  <. x ,  y >.  e.  <. u ,  t >. )
)
38 opeq12 3580 . . . . . . . . . . . . . 14  |-  ( ( z  =  x  /\  w  =  x )  -> 
<. z ,  w >.  = 
<. x ,  x >. )
39383adant3 959 . . . . . . . . . . . . 13  |-  ( ( z  =  x  /\  w  =  x  /\  v  =  y )  -> 
<. z ,  w >.  = 
<. x ,  x >. )
4039eleq1d 2148 . . . . . . . . . . . 12  |-  ( ( z  =  x  /\  w  =  x  /\  v  =  y )  ->  ( <. z ,  w >.  e.  <. u ,  t
>. 
<-> 
<. x ,  x >.  e. 
<. u ,  t >.
) )
41 opeq12 3580 . . . . . . . . . . . . . 14  |-  ( ( z  =  x  /\  v  =  y )  -> 
<. z ,  v >.  =  <. x ,  y
>. )
42413adant2 958 . . . . . . . . . . . . 13  |-  ( ( z  =  x  /\  w  =  x  /\  v  =  y )  -> 
<. z ,  v >.  =  <. x ,  y
>. )
4342eleq1d 2148 . . . . . . . . . . . 12  |-  ( ( z  =  x  /\  w  =  x  /\  v  =  y )  ->  ( <. z ,  v
>.  e.  <. u ,  t
>. 
<-> 
<. x ,  y >.  e.  <. u ,  t
>. ) )
4440, 43anbi12d 457 . . . . . . . . . . 11  |-  ( ( z  =  x  /\  w  =  x  /\  v  =  y )  ->  ( ( <. z ,  w >.  e.  <. u ,  t >.  /\  <. z ,  v >.  e.  <. u ,  t >. )  <->  (
<. x ,  x >.  e. 
<. u ,  t >.  /\  <. x ,  y
>.  e.  <. u ,  t
>. ) ) )
45 eqeq12 2094 . . . . . . . . . . . 12  |-  ( ( w  =  x  /\  v  =  y )  ->  ( w  =  v  <-> 
x  =  y ) )
46453adant1 957 . . . . . . . . . . 11  |-  ( ( z  =  x  /\  w  =  x  /\  v  =  y )  ->  ( w  =  v  <-> 
x  =  y ) )
4744, 46imbi12d 232 . . . . . . . . . 10  |-  ( ( z  =  x  /\  w  =  x  /\  v  =  y )  ->  ( ( ( <.
z ,  w >.  e. 
<. u ,  t >.  /\  <. z ,  v
>.  e.  <. u ,  t
>. )  ->  w  =  v )  <->  ( ( <. x ,  x >.  e. 
<. u ,  t >.  /\  <. x ,  y
>.  e.  <. u ,  t
>. )  ->  x  =  y ) ) )
4847spc3gv 2691 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  x  e.  _V  /\  y  e.  _V )  ->  ( A. z A. w A. v ( ( <.
z ,  w >.  e. 
<. u ,  t >.  /\  <. z ,  v
>.  e.  <. u ,  t
>. )  ->  w  =  v )  ->  (
( <. x ,  x >.  e.  <. u ,  t
>.  /\  <. x ,  y
>.  e.  <. u ,  t
>. )  ->  x  =  y ) ) )
4915, 15, 18, 48mp3an 1269 . . . . . . . 8  |-  ( A. z A. w A. v
( ( <. z ,  w >.  e.  <. u ,  t >.  /\  <. z ,  v >.  e.  <. u ,  t >. )  ->  w  =  v )  ->  ( ( <.
x ,  x >.  e. 
<. u ,  t >.  /\  <. x ,  y
>.  e.  <. u ,  t
>. )  ->  x  =  y ) )
5028, 37, 49syl2im 38 . . . . . . 7  |-  ( Fun 
<. u ,  t >.  ->  ( <. u ,  t
>.  =  { <. x ,  x >. ,  <. x ,  y >. }  ->  x  =  y ) )
5126, 50syl5bi 150 . . . . . 6  |-  ( Fun 
<. u ,  t >.  ->  ( ( u  =  { x }  /\  t  =  { x ,  y } )  ->  x  =  y ) )
52 dfsn2 3420 . . . . . . . . . . 11  |-  { x }  =  { x ,  x }
53 preq2 3478 . . . . . . . . . . 11  |-  ( x  =  y  ->  { x ,  x }  =  {
x ,  y } )
5452, 53syl5req 2127 . . . . . . . . . 10  |-  ( x  =  y  ->  { x ,  y }  =  { x } )
5554eqeq2d 2093 . . . . . . . . 9  |-  ( x  =  y  ->  (
t  =  { x ,  y }  <->  t  =  { x } ) )
56 eqtr3 2101 . . . . . . . . . 10  |-  ( ( u  =  { x }  /\  t  =  {
x } )  ->  u  =  t )
5756expcom 114 . . . . . . . . 9  |-  ( t  =  { x }  ->  ( u  =  {
x }  ->  u  =  t ) )
5855, 57syl6bi 161 . . . . . . . 8  |-  ( x  =  y  ->  (
t  =  { x ,  y }  ->  ( u  =  { x }  ->  u  =  t ) ) )
5958com13 79 . . . . . . 7  |-  ( u  =  { x }  ->  ( t  =  {
x ,  y }  ->  ( x  =  y  ->  u  =  t ) ) )
6059imp 122 . . . . . 6  |-  ( ( u  =  { x }  /\  t  =  {
x ,  y } )  ->  ( x  =  y  ->  u  =  t ) )
6151, 60sylcom 28 . . . . 5  |-  ( Fun 
<. u ,  t >.  ->  ( ( u  =  { x }  /\  t  =  { x ,  y } )  ->  u  =  t ) )
6261exlimdvv 1819 . . . 4  |-  ( Fun 
<. u ,  t >.  ->  ( E. x E. y ( u  =  { x }  /\  t  =  { x ,  y } )  ->  u  =  t ) )
6313, 62mpd 13 . . 3  |-  ( Fun 
<. u ,  t >.  ->  u  =  t )
644, 8, 63vtocl2g 2663 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( Fun  <. A ,  B >.  ->  A  =  B ) )
65643impia 1136 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  Fun  <. A ,  B >. )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920   A.wal 1283    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2602   {csn 3406   {cpr 3407   <.cop 3409   Rel wrel 4376   Fun wfun 4926
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-fun 4934
This theorem is referenced by: (None)
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