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Theorem fundmen 6503
Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypothesis
Ref Expression
fundmen.1  |-  F  e. 
_V
Assertion
Ref Expression
fundmen  |-  ( Fun 
F  ->  dom  F  ~~  F )

Proof of Theorem fundmen
Dummy variables  x  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fundmen.1 . . . 4  |-  F  e. 
_V
21dmex 4687 . . 3  |-  dom  F  e.  _V
32a1i 9 . 2  |-  ( Fun 
F  ->  dom  F  e. 
_V )
41a1i 9 . 2  |-  ( Fun 
F  ->  F  e.  _V )
5 funfvop 5395 . . 3  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  <. x ,  ( F `
 x ) >.  e.  F )
65ex 113 . 2  |-  ( Fun 
F  ->  ( x  e.  dom  F  ->  <. x ,  ( F `  x ) >.  e.  F
) )
7 funrel 5019 . . 3  |-  ( Fun 
F  ->  Rel  F )
8 elreldm 4649 . . . 4  |-  ( ( Rel  F  /\  y  e.  F )  ->  |^| |^| y  e.  dom  F )
98ex 113 . . 3  |-  ( Rel 
F  ->  ( y  e.  F  ->  |^| |^| y  e.  dom  F ) )
107, 9syl 14 . 2  |-  ( Fun 
F  ->  ( y  e.  F  ->  |^| |^| y  e.  dom  F ) )
11 df-rel 4435 . . . . . . . . 9  |-  ( Rel 
F  <->  F  C_  ( _V 
X.  _V ) )
127, 11sylib 120 . . . . . . . 8  |-  ( Fun 
F  ->  F  C_  ( _V  X.  _V ) )
1312sselda 3023 . . . . . . 7  |-  ( ( Fun  F  /\  y  e.  F )  ->  y  e.  ( _V  X.  _V ) )
14 elvv 4488 . . . . . . 7  |-  ( y  e.  ( _V  X.  _V )  <->  E. z E. w  y  =  <. z ,  w >. )
1513, 14sylib 120 . . . . . 6  |-  ( ( Fun  F  /\  y  e.  F )  ->  E. z E. w  y  =  <. z ,  w >. )
16 inteq 3686 . . . . . . . . . . . . . . . . 17  |-  ( y  =  <. z ,  w >.  ->  |^| y  =  |^| <.
z ,  w >. )
1716inteqd 3688 . . . . . . . . . . . . . . . 16  |-  ( y  =  <. z ,  w >.  ->  |^| |^| y  =  |^| |^|
<. z ,  w >. )
18 vex 2622 . . . . . . . . . . . . . . . . 17  |-  z  e. 
_V
19 vex 2622 . . . . . . . . . . . . . . . . 17  |-  w  e. 
_V
2018, 19op1stb 4290 . . . . . . . . . . . . . . . 16  |-  |^| |^| <. z ,  w >.  =  z
2117, 20syl6eq 2136 . . . . . . . . . . . . . . 15  |-  ( y  =  <. z ,  w >.  ->  |^| |^| y  =  z )
22 eqeq1 2094 . . . . . . . . . . . . . . 15  |-  ( x  =  |^| |^| y  ->  ( x  =  z  <->  |^| |^| y  =  z ) )
2321, 22syl5ibr 154 . . . . . . . . . . . . . 14  |-  ( x  =  |^| |^| y  ->  ( y  =  <. z ,  w >.  ->  x  =  z ) )
24 opeq1 3617 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  <. x ,  w >.  =  <. z ,  w >. )
2523, 24syl6 33 . . . . . . . . . . . . 13  |-  ( x  =  |^| |^| y  ->  ( y  =  <. z ,  w >.  ->  <. x ,  w >.  =  <. z ,  w >. )
)
2625imp 122 . . . . . . . . . . . 12  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  -> 
<. x ,  w >.  = 
<. z ,  w >. )
27 eqeq2 2097 . . . . . . . . . . . . . 14  |-  ( <.
x ,  w >.  = 
<. z ,  w >.  -> 
( y  =  <. x ,  w >.  <->  y  =  <. z ,  w >. ) )
2827biimprcd 158 . . . . . . . . . . . . 13  |-  ( y  =  <. z ,  w >.  ->  ( <. x ,  w >.  =  <. z ,  w >.  ->  y  =  <. x ,  w >. ) )
2928adantl 271 . . . . . . . . . . . 12  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  ->  ( <. x ,  w >.  =  <. z ,  w >.  ->  y  =  <. x ,  w >. )
)
3026, 29mpd 13 . . . . . . . . . . 11  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  ->  y  =  <. x ,  w >. )
3130ancoms 264 . . . . . . . . . 10  |-  ( ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y )  -> 
y  =  <. x ,  w >. )
3231adantl 271 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  y  =  <. x ,  w >. )
3330eleq1d 2156 . . . . . . . . . . . . . . 15  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  ->  ( y  e.  F  <->  <.
x ,  w >.  e.  F ) )
3433adantl 271 . . . . . . . . . . . . . 14  |-  ( ( Fun  F  /\  (
x  =  |^| |^| y  /\  y  =  <. z ,  w >. )
)  ->  ( y  e.  F  <->  <. x ,  w >.  e.  F ) )
35 funopfv 5328 . . . . . . . . . . . . . . 15  |-  ( Fun 
F  ->  ( <. x ,  w >.  e.  F  ->  ( F `  x
)  =  w ) )
3635adantr 270 . . . . . . . . . . . . . 14  |-  ( ( Fun  F  /\  (
x  =  |^| |^| y  /\  y  =  <. z ,  w >. )
)  ->  ( <. x ,  w >.  e.  F  ->  ( F `  x
)  =  w ) )
3734, 36sylbid 148 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\  (
x  =  |^| |^| y  /\  y  =  <. z ,  w >. )
)  ->  ( y  e.  F  ->  ( F `
 x )  =  w ) )
3837exp32 357 . . . . . . . . . . . 12  |-  ( Fun 
F  ->  ( x  =  |^| |^| y  ->  (
y  =  <. z ,  w >.  ->  ( y  e.  F  ->  ( F `  x )  =  w ) ) ) )
3938com24 86 . . . . . . . . . . 11  |-  ( Fun 
F  ->  ( y  e.  F  ->  ( y  =  <. z ,  w >.  ->  ( x  = 
|^| |^| y  ->  ( F `  x )  =  w ) ) ) )
4039imp43 347 . . . . . . . . . 10  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  ( F `  x )  =  w )
4140opeq2d 3624 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  <. x ,  ( F `  x )
>.  =  <. x ,  w >. )
4232, 41eqtr4d 2123 . . . . . . . 8  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  y  =  <. x ,  ( F `  x ) >. )
4342exp32 357 . . . . . . 7  |-  ( ( Fun  F  /\  y  e.  F )  ->  (
y  =  <. z ,  w >.  ->  ( x  =  |^| |^| y  ->  y  =  <. x ,  ( F `  x ) >. )
) )
4443exlimdvv 1825 . . . . . 6  |-  ( ( Fun  F  /\  y  e.  F )  ->  ( E. z E. w  y  =  <. z ,  w >.  ->  ( x  = 
|^| |^| y  ->  y  =  <. x ,  ( F `  x )
>. ) ) )
4515, 44mpd 13 . . . . 5  |-  ( ( Fun  F  /\  y  e.  F )  ->  (
x  =  |^| |^| y  ->  y  =  <. x ,  ( F `  x ) >. )
)
4645adantrl 462 . . . 4  |-  ( ( Fun  F  /\  (
x  e.  dom  F  /\  y  e.  F
) )  ->  (
x  =  |^| |^| y  ->  y  =  <. x ,  ( F `  x ) >. )
)
47 inteq 3686 . . . . . . . . 9  |-  ( y  =  <. x ,  ( F `  x )
>.  ->  |^| y  =  |^| <.
x ,  ( F `
 x ) >.
)
4847inteqd 3688 . . . . . . . 8  |-  ( y  =  <. x ,  ( F `  x )
>.  ->  |^| |^| y  =  |^| |^|
<. x ,  ( F `
 x ) >.
)
4948adantl 271 . . . . . . 7  |-  ( ( ( Fun  F  /\  x  e.  dom  F )  /\  y  =  <. x ,  ( F `  x ) >. )  ->  |^| |^| y  =  |^| |^|
<. x ,  ( F `
 x ) >.
)
50 vex 2622 . . . . . . . . 9  |-  x  e. 
_V
51 funfvex 5306 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
52 op1stbg 4291 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  ( F `  x )  e.  _V )  ->  |^| |^| <. x ,  ( F `  x )
>.  =  x )
5350, 51, 52sylancr 405 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  |^| |^| <. x ,  ( F `  x )
>.  =  x )
5453adantr 270 . . . . . . 7  |-  ( ( ( Fun  F  /\  x  e.  dom  F )  /\  y  =  <. x ,  ( F `  x ) >. )  ->  |^| |^| <. x ,  ( F `  x )
>.  =  x )
5549, 54eqtr2d 2121 . . . . . 6  |-  ( ( ( Fun  F  /\  x  e.  dom  F )  /\  y  =  <. x ,  ( F `  x ) >. )  ->  x  =  |^| |^| y
)
5655ex 113 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( y  =  <. x ,  ( F `  x ) >.  ->  x  =  |^| |^| y ) )
5756adantrr 463 . . . 4  |-  ( ( Fun  F  /\  (
x  e.  dom  F  /\  y  e.  F
) )  ->  (
y  =  <. x ,  ( F `  x ) >.  ->  x  =  |^| |^| y ) )
5846, 57impbid 127 . . 3  |-  ( ( Fun  F  /\  (
x  e.  dom  F  /\  y  e.  F
) )  ->  (
x  =  |^| |^| y  <->  y  =  <. x ,  ( F `  x )
>. ) )
5958ex 113 . 2  |-  ( Fun 
F  ->  ( (
x  e.  dom  F  /\  y  e.  F
)  ->  ( x  =  |^| |^| y  <->  y  =  <. x ,  ( F `
 x ) >.
) ) )
603, 4, 6, 10, 59en3d 6466 1  |-  ( Fun 
F  ->  dom  F  ~~  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619    C_ wss 2997   <.cop 3444   |^|cint 3683   class class class wbr 3837    X. cxp 4426   dom cdm 4428   Rel wrel 4433   Fun wfun 4996   ` cfv 5002    ~~ cen 6435
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-en 6438
This theorem is referenced by:  fundmeng  6504
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