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Theorem fundmen 6819
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
StepHypRef Expression
1 fundmen.1 . . . 4  |-  F  e. 
_V
21dmex 4848 . . 3  |-  dom  F  e.  _V
32a1i 12 . 2  |-  ( Fun 
F  ->  dom  F  e. 
_V )
41a1i 12 . 2  |-  ( Fun 
F  ->  F  e.  _V )
5 funfvop 5489 . . 3  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  <. x ,  ( F `
 x ) >.  e.  F )
65ex 425 . 2  |-  ( Fun 
F  ->  ( x  e.  dom  F  ->  <. x ,  ( F `  x ) >.  e.  F
) )
7 funrel 5130 . . 3  |-  ( Fun 
F  ->  Rel  F )
8 elreldm 4810 . . . 4  |-  ( ( Rel  F  /\  y  e.  F )  ->  |^| |^| y  e.  dom  F )
98ex 425 . . 3  |-  ( Rel 
F  ->  ( y  e.  F  ->  |^| |^| y  e.  dom  F ) )
107, 9syl 17 . 2  |-  ( Fun 
F  ->  ( y  e.  F  ->  |^| |^| y  e.  dom  F ) )
11 df-rel 4595 . . . . . . . . 9  |-  ( Rel 
F  <->  F  C_  ( _V 
X.  _V ) )
127, 11sylib 190 . . . . . . . 8  |-  ( Fun 
F  ->  F  C_  ( _V  X.  _V ) )
1312sselda 3103 . . . . . . 7  |-  ( ( Fun  F  /\  y  e.  F )  ->  y  e.  ( _V  X.  _V ) )
14 elvv 4655 . . . . . . 7  |-  ( y  e.  ( _V  X.  _V )  <->  E. z E. w  y  =  <. z ,  w >. )
1513, 14sylib 190 . . . . . 6  |-  ( ( Fun  F  /\  y  e.  F )  ->  E. z E. w  y  =  <. z ,  w >. )
16 inteq 3763 . . . . . . . . . . . . . . . . 17  |-  ( y  =  <. z ,  w >.  ->  |^| y  =  |^| <.
z ,  w >. )
1716inteqd 3765 . . . . . . . . . . . . . . . 16  |-  ( y  =  <. z ,  w >.  ->  |^| |^| y  =  |^| |^|
<. z ,  w >. )
18 vex 2730 . . . . . . . . . . . . . . . . 17  |-  z  e. 
_V
19 vex 2730 . . . . . . . . . . . . . . . . 17  |-  w  e. 
_V
2018, 19op1stb 4460 . . . . . . . . . . . . . . . 16  |-  |^| |^| <. z ,  w >.  =  z
2117, 20syl6eq 2301 . . . . . . . . . . . . . . 15  |-  ( y  =  <. z ,  w >.  ->  |^| |^| y  =  z )
22 eqeq1 2259 . . . . . . . . . . . . . . 15  |-  ( x  =  |^| |^| y  ->  ( x  =  z  <->  |^| |^| y  =  z ) )
2321, 22syl5ibr 214 . . . . . . . . . . . . . 14  |-  ( x  =  |^| |^| y  ->  ( y  =  <. z ,  w >.  ->  x  =  z ) )
24 opeq1 3696 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  <. x ,  w >.  =  <. z ,  w >. )
2523, 24syl6 31 . . . . . . . . . . . . 13  |-  ( x  =  |^| |^| y  ->  ( y  =  <. z ,  w >.  ->  <. x ,  w >.  =  <. z ,  w >. )
)
2625imp 420 . . . . . . . . . . . 12  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  -> 
<. x ,  w >.  = 
<. z ,  w >. )
27 eqeq2 2262 . . . . . . . . . . . . . 14  |-  ( <.
x ,  w >.  = 
<. z ,  w >.  -> 
( y  =  <. x ,  w >.  <->  y  =  <. z ,  w >. ) )
2827biimprcd 218 . . . . . . . . . . . . 13  |-  ( y  =  <. z ,  w >.  ->  ( <. x ,  w >.  =  <. z ,  w >.  ->  y  =  <. x ,  w >. ) )
2928adantl 454 . . . . . . . . . . . 12  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  ->  ( <. x ,  w >.  =  <. z ,  w >.  ->  y  =  <. x ,  w >. )
)
3026, 29mpd 16 . . . . . . . . . . 11  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  ->  y  =  <. x ,  w >. )
3130ancoms 441 . . . . . . . . . 10  |-  ( ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y )  -> 
y  =  <. x ,  w >. )
3231adantl 454 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  y  =  <. x ,  w >. )
3330eleq1d 2319 . . . . . . . . . . . . . . 15  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  ->  ( y  e.  F  <->  <.
x ,  w >.  e.  F ) )
3433adantl 454 . . . . . . . . . . . . . 14  |-  ( ( Fun  F  /\  (
x  =  |^| |^| y  /\  y  =  <. z ,  w >. )
)  ->  ( y  e.  F  <->  <. x ,  w >.  e.  F ) )
35 funopfv 5414 . . . . . . . . . . . . . . 15  |-  ( Fun 
F  ->  ( <. x ,  w >.  e.  F  ->  ( F `  x
)  =  w ) )
3635adantr 453 . . . . . . . . . . . . . 14  |-  ( ( Fun  F  /\  (
x  =  |^| |^| y  /\  y  =  <. z ,  w >. )
)  ->  ( <. x ,  w >.  e.  F  ->  ( F `  x
)  =  w ) )
3734, 36sylbid 208 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\  (
x  =  |^| |^| y  /\  y  =  <. z ,  w >. )
)  ->  ( y  e.  F  ->  ( F `
 x )  =  w ) )
3837exp32 591 . . . . . . . . . . . 12  |-  ( Fun 
F  ->  ( x  =  |^| |^| y  ->  (
y  =  <. z ,  w >.  ->  ( y  e.  F  ->  ( F `  x )  =  w ) ) ) )
3938com24 83 . . . . . . . . . . 11  |-  ( Fun 
F  ->  ( y  e.  F  ->  ( y  =  <. z ,  w >.  ->  ( x  = 
|^| |^| y  ->  ( F `  x )  =  w ) ) ) )
4039imp43 581 . . . . . . . . . 10  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  ( F `  x )  =  w )
4140opeq2d 3703 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  <. x ,  ( F `  x )
>.  =  <. x ,  w >. )
4232, 41eqtr4d 2288 . . . . . . . 8  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  y  =  <. x ,  ( F `  x ) >. )
4342exp32 591 . . . . . . 7  |-  ( ( Fun  F  /\  y  e.  F )  ->  (
y  =  <. z ,  w >.  ->  ( x  =  |^| |^| y  ->  y  =  <. x ,  ( F `  x ) >. )
) )
4443exlimdvv 2026 . . . . . 6  |-  ( ( Fun  F  /\  y  e.  F )  ->  ( E. z E. w  y  =  <. z ,  w >.  ->  ( x  = 
|^| |^| y  ->  y  =  <. x ,  ( F `  x )
>. ) ) )
4515, 44mpd 16 . . . . 5  |-  ( ( Fun  F  /\  y  e.  F )  ->  (
x  =  |^| |^| y  ->  y  =  <. x ,  ( F `  x ) >. )
)
4645adantrl 699 . . . 4  |-  ( ( Fun  F  /\  (
x  e.  dom  F  /\  y  e.  F
) )  ->  (
x  =  |^| |^| y  ->  y  =  <. x ,  ( F `  x ) >. )
)
47 inteq 3763 . . . . . 6  |-  ( y  =  <. x ,  ( F `  x )
>.  ->  |^| y  =  |^| <.
x ,  ( F `
 x ) >.
)
4847inteqd 3765 . . . . 5  |-  ( y  =  <. x ,  ( F `  x )
>.  ->  |^| |^| y  =  |^| |^|
<. x ,  ( F `
 x ) >.
)
49 vex 2730 . . . . . 6  |-  x  e. 
_V
50 fvex 5391 . . . . . 6  |-  ( F `
 x )  e. 
_V
5149, 50op1stb 4460 . . . . 5  |-  |^| |^| <. x ,  ( F `  x ) >.  =  x
5248, 51syl6req 2302 . . . 4  |-  ( y  =  <. x ,  ( F `  x )
>.  ->  x  =  |^| |^| y )
5346, 52impbid1 196 . . 3  |-  ( ( Fun  F  /\  (
x  e.  dom  F  /\  y  e.  F
) )  ->  (
x  =  |^| |^| y  <->  y  =  <. x ,  ( F `  x )
>. ) )
5453ex 425 . 2  |-  ( Fun 
F  ->  ( (
x  e.  dom  F  /\  y  e.  F
)  ->  ( x  =  |^| |^| y  <->  y  =  <. x ,  ( F `
 x ) >.
) ) )
553, 4, 6, 10, 54en3d 6784 1  |-  ( Fun 
F  ->  dom  F  ~~  F )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   _Vcvv 2727    C_ wss 3078   <.cop 3547   |^|cint 3760   class class class wbr 3920    X. cxp 4578   dom cdm 4580   Rel wrel 4585   Fun wfun 4586   ` cfv 4592    ~~ cen 6746
This theorem is referenced by:  fundmeng  6820  infmap2  7728  fnctartar  25073  fnctartar2  25074
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-en 6750
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