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Theorem fundmen 6977
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 4978 . . 3  |-  dom  F  e.  _V
32a1i 10 . 2  |-  ( Fun 
F  ->  dom  F  e. 
_V )
41a1i 10 . 2  |-  ( Fun 
F  ->  F  e.  _V )
5 funfvop 5675 . . 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 5309 . . 3  |-  ( Fun 
F  ->  Rel  F )
8 elreldm 4940 . . . 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 4733 . . . . . . . . 9  |-  ( Rel 
F  <->  F  C_  ( _V 
X.  _V ) )
127, 11sylib 188 . . . . . . . 8  |-  ( Fun 
F  ->  F  C_  ( _V  X.  _V ) )
1312sselda 3214 . . . . . . 7  |-  ( ( Fun  F  /\  y  e.  F )  ->  y  e.  ( _V  X.  _V ) )
14 elvv 4785 . . . . . . 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 3902 . . . . . . . . . . . . . . . . 17  |-  ( y  =  <. z ,  w >.  ->  |^| y  =  |^| <.
z ,  w >. )
1716inteqd 3904 . . . . . . . . . . . . . . . 16  |-  ( y  =  <. z ,  w >.  ->  |^| |^| y  =  |^| |^|
<. z ,  w >. )
18 vex 2825 . . . . . . . . . . . . . . . . 17  |-  z  e. 
_V
19 vex 2825 . . . . . . . . . . . . . . . . 17  |-  w  e. 
_V
2018, 19op1stb 4606 . . . . . . . . . . . . . . . 16  |-  |^| |^| <. z ,  w >.  =  z
2117, 20syl6eq 2364 . . . . . . . . . . . . . . 15  |-  ( y  =  <. z ,  w >.  ->  |^| |^| y  =  z )
22 eqeq1 2322 . . . . . . . . . . . . . . 15  |-  ( x  =  |^| |^| y  ->  ( x  =  z  <->  |^| |^| y  =  z ) )
2321, 22syl5ibr 212 . . . . . . . . . . . . . 14  |-  ( x  =  |^| |^| y  ->  ( y  =  <. z ,  w >.  ->  x  =  z ) )
24 opeq1 3833 . . . . . . . . . . . . . 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 2325 . . . . . . . . . . . . . 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 2382 . . . . . . . . . . . . . . 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 5600 . . . . . . . . . . . . . . 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 3840 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  <. x ,  ( F `  x )
>.  =  <. x ,  w >. )
4232, 41eqtr4d 2351 . . . . . . . 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 1628 . . . . . 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 3902 . . . . . 6  |-  ( y  =  <. x ,  ( F `  x )
>.  ->  |^| y  =  |^| <.
x ,  ( F `
 x ) >.
)
4847inteqd 3904 . . . . 5  |-  ( y  =  <. x ,  ( F `  x )
>.  ->  |^| |^| y  =  |^| |^|
<. x ,  ( F `
 x ) >.
)
49 vex 2825 . . . . . 6  |-  x  e. 
_V
50 fvex 5577 . . . . . 6  |-  ( F `
 x )  e. 
_V
5149, 50op1stb 4606 . . . . 5  |-  |^| |^| <. x ,  ( F `  x ) >.  =  x
5248, 51syl6req 2365 . . . 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 6941 1  |-  ( Fun 
F  ->  dom  F  ~~  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1532    = wceq 1633    e. wcel 1701   _Vcvv 2822    C_ wss 3186   <.cop 3677   |^|cint 3899   class class class wbr 4060    X. cxp 4724   dom cdm 4726   Rel wrel 4731   Fun wfun 5286   ` cfv 5292    ~~ cen 6903
This theorem is referenced by:  fundmeng  6978  infmap2  7889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-int 3900  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-en 6907
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