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Theorem fundmen 6934
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 4941 . . 3  |-  dom  F  e.  _V
32a1i 10 . 2  |-  ( Fun 
F  ->  dom  F  e. 
_V )
41a1i 10 . 2  |-  ( Fun 
F  ->  F  e.  _V )
5 funfvop 5637 . . 3  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  <. x ,  ( F `
 x ) >.  e.  F )
65ex 423 . 2  |-  ( Fun 
F  ->  ( x  e.  dom  F  ->  <. x ,  ( F `  x ) >.  e.  F
) )
7 funrel 5272 . . 3  |-  ( Fun 
F  ->  Rel  F )
8 elreldm 4903 . . . 4  |-  ( ( Rel  F  /\  y  e.  F )  ->  |^| |^| y  e.  dom  F )
98ex 423 . . 3  |-  ( Rel 
F  ->  ( y  e.  F  ->  |^| |^| y  e.  dom  F ) )
107, 9syl 15 . 2  |-  ( Fun 
F  ->  ( y  e.  F  ->  |^| |^| y  e.  dom  F ) )
11 df-rel 4696 . . . . . . . . 9  |-  ( Rel 
F  <->  F  C_  ( _V 
X.  _V ) )
127, 11sylib 188 . . . . . . . 8  |-  ( Fun 
F  ->  F  C_  ( _V  X.  _V ) )
1312sselda 3180 . . . . . . 7  |-  ( ( Fun  F  /\  y  e.  F )  ->  y  e.  ( _V  X.  _V ) )
14 elvv 4748 . . . . . . 7  |-  ( y  e.  ( _V  X.  _V )  <->  E. z E. w  y  =  <. z ,  w >. )
1513, 14sylib 188 . . . . . 6  |-  ( ( Fun  F  /\  y  e.  F )  ->  E. z E. w  y  =  <. z ,  w >. )
16 inteq 3865 . . . . . . . . . . . . . . . . 17  |-  ( y  =  <. z ,  w >.  ->  |^| y  =  |^| <.
z ,  w >. )
1716inteqd 3867 . . . . . . . . . . . . . . . 16  |-  ( y  =  <. z ,  w >.  ->  |^| |^| y  =  |^| |^|
<. z ,  w >. )
18 vex 2791 . . . . . . . . . . . . . . . . 17  |-  z  e. 
_V
19 vex 2791 . . . . . . . . . . . . . . . . 17  |-  w  e. 
_V
2018, 19op1stb 4569 . . . . . . . . . . . . . . . 16  |-  |^| |^| <. z ,  w >.  =  z
2117, 20syl6eq 2331 . . . . . . . . . . . . . . 15  |-  ( y  =  <. z ,  w >.  ->  |^| |^| y  =  z )
22 eqeq1 2289 . . . . . . . . . . . . . . 15  |-  ( x  =  |^| |^| y  ->  ( x  =  z  <->  |^| |^| y  =  z ) )
2321, 22syl5ibr 212 . . . . . . . . . . . . . 14  |-  ( x  =  |^| |^| y  ->  ( y  =  <. z ,  w >.  ->  x  =  z ) )
24 opeq1 3796 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  <. x ,  w >.  =  <. z ,  w >. )
2523, 24syl6 29 . . . . . . . . . . . . 13  |-  ( x  =  |^| |^| y  ->  ( y  =  <. z ,  w >.  ->  <. x ,  w >.  =  <. z ,  w >. )
)
2625imp 418 . . . . . . . . . . . 12  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  -> 
<. x ,  w >.  = 
<. z ,  w >. )
27 eqeq2 2292 . . . . . . . . . . . . . 14  |-  ( <.
x ,  w >.  = 
<. z ,  w >.  -> 
( y  =  <. x ,  w >.  <->  y  =  <. z ,  w >. ) )
2827biimprcd 216 . . . . . . . . . . . . 13  |-  ( y  =  <. z ,  w >.  ->  ( <. x ,  w >.  =  <. z ,  w >.  ->  y  =  <. x ,  w >. ) )
2928adantl 452 . . . . . . . . . . . 12  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  ->  ( <. x ,  w >.  =  <. z ,  w >.  ->  y  =  <. x ,  w >. )
)
3026, 29mpd 14 . . . . . . . . . . 11  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  ->  y  =  <. x ,  w >. )
3130ancoms 439 . . . . . . . . . 10  |-  ( ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y )  -> 
y  =  <. x ,  w >. )
3231adantl 452 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  y  =  <. x ,  w >. )
3330eleq1d 2349 . . . . . . . . . . . . . . 15  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  ->  ( y  e.  F  <->  <.
x ,  w >.  e.  F ) )
3433adantl 452 . . . . . . . . . . . . . 14  |-  ( ( Fun  F  /\  (
x  =  |^| |^| y  /\  y  =  <. z ,  w >. )
)  ->  ( y  e.  F  <->  <. x ,  w >.  e.  F ) )
35 funopfv 5562 . . . . . . . . . . . . . . 15  |-  ( Fun 
F  ->  ( <. x ,  w >.  e.  F  ->  ( F `  x
)  =  w ) )
3635adantr 451 . . . . . . . . . . . . . 14  |-  ( ( Fun  F  /\  (
x  =  |^| |^| y  /\  y  =  <. z ,  w >. )
)  ->  ( <. x ,  w >.  e.  F  ->  ( F `  x
)  =  w ) )
3734, 36sylbid 206 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\  (
x  =  |^| |^| y  /\  y  =  <. z ,  w >. )
)  ->  ( y  e.  F  ->  ( F `
 x )  =  w ) )
3837exp32 588 . . . . . . . . . . . 12  |-  ( Fun 
F  ->  ( x  =  |^| |^| y  ->  (
y  =  <. z ,  w >.  ->  ( y  e.  F  ->  ( F `  x )  =  w ) ) ) )
3938com24 81 . . . . . . . . . . 11  |-  ( Fun 
F  ->  ( y  e.  F  ->  ( y  =  <. z ,  w >.  ->  ( x  = 
|^| |^| y  ->  ( F `  x )  =  w ) ) ) )
4039imp43 578 . . . . . . . . . 10  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  ( F `  x )  =  w )
4140opeq2d 3803 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  <. x ,  ( F `  x )
>.  =  <. x ,  w >. )
4232, 41eqtr4d 2318 . . . . . . . 8  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  y  =  <. x ,  ( F `  x ) >. )
4342exp32 588 . . . . . . 7  |-  ( ( Fun  F  /\  y  e.  F )  ->  (
y  =  <. z ,  w >.  ->  ( x  =  |^| |^| y  ->  y  =  <. x ,  ( F `  x ) >. )
) )
4443exlimdvv 1668 . . . . . 6  |-  ( ( Fun  F  /\  y  e.  F )  ->  ( E. z E. w  y  =  <. z ,  w >.  ->  ( x  = 
|^| |^| y  ->  y  =  <. x ,  ( F `  x )
>. ) ) )
4515, 44mpd 14 . . . . 5  |-  ( ( Fun  F  /\  y  e.  F )  ->  (
x  =  |^| |^| y  ->  y  =  <. x ,  ( F `  x ) >. )
)
4645adantrl 696 . . . 4  |-  ( ( Fun  F  /\  (
x  e.  dom  F  /\  y  e.  F
) )  ->  (
x  =  |^| |^| y  ->  y  =  <. x ,  ( F `  x ) >. )
)
47 inteq 3865 . . . . . 6  |-  ( y  =  <. x ,  ( F `  x )
>.  ->  |^| y  =  |^| <.
x ,  ( F `
 x ) >.
)
4847inteqd 3867 . . . . 5  |-  ( y  =  <. x ,  ( F `  x )
>.  ->  |^| |^| y  =  |^| |^|
<. x ,  ( F `
 x ) >.
)
49 vex 2791 . . . . . 6  |-  x  e. 
_V
50 fvex 5539 . . . . . 6  |-  ( F `
 x )  e. 
_V
5149, 50op1stb 4569 . . . . 5  |-  |^| |^| <. x ,  ( F `  x ) >.  =  x
5248, 51syl6req 2332 . . . 4  |-  ( y  =  <. x ,  ( F `  x )
>.  ->  x  =  |^| |^| y )
5346, 52impbid1 194 . . 3  |-  ( ( Fun  F  /\  (
x  e.  dom  F  /\  y  e.  F
) )  ->  (
x  =  |^| |^| y  <->  y  =  <. x ,  ( F `  x )
>. ) )
5453ex 423 . 2  |-  ( Fun 
F  ->  ( (
x  e.  dom  F  /\  y  e.  F
)  ->  ( x  =  |^| |^| y  <->  y  =  <. x ,  ( F `
 x ) >.
) ) )
553, 4, 6, 10, 54en3d 6898 1  |-  ( Fun 
F  ->  dom  F  ~~  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   <.cop 3643   |^|cint 3862   class class class wbr 4023    X. cxp 4687   dom cdm 4689   Rel wrel 4694   Fun wfun 5249   ` cfv 5255    ~~ cen 6860
This theorem is referenced by:  fundmeng  6935  infmap2  7844  fnctartar  25907  fnctartar2  25908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-en 6864
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