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Theorem fundmen 7496
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 6624 . . 3  |-  dom  F  e.  _V
32a1i 11 . 2  |-  ( Fun 
F  ->  dom  F  e. 
_V )
41a1i 11 . 2  |-  ( Fun 
F  ->  F  e.  _V )
5 funfvop 5927 . . 3  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  <. x ,  ( F `
 x ) >.  e.  F )
65ex 434 . 2  |-  ( Fun 
F  ->  ( x  e.  dom  F  ->  <. x ,  ( F `  x ) >.  e.  F
) )
7 funrel 5546 . . 3  |-  ( Fun 
F  ->  Rel  F )
8 elreldm 5175 . . . 4  |-  ( ( Rel  F  /\  y  e.  F )  ->  |^| |^| y  e.  dom  F )
98ex 434 . . 3  |-  ( Rel 
F  ->  ( y  e.  F  ->  |^| |^| y  e.  dom  F ) )
107, 9syl 16 . 2  |-  ( Fun 
F  ->  ( y  e.  F  ->  |^| |^| y  e.  dom  F ) )
11 df-rel 4958 . . . . . . . . 9  |-  ( Rel 
F  <->  F  C_  ( _V 
X.  _V ) )
127, 11sylib 196 . . . . . . . 8  |-  ( Fun 
F  ->  F  C_  ( _V  X.  _V ) )
1312sselda 3467 . . . . . . 7  |-  ( ( Fun  F  /\  y  e.  F )  ->  y  e.  ( _V  X.  _V ) )
14 elvv 5008 . . . . . . 7  |-  ( y  e.  ( _V  X.  _V )  <->  E. z E. w  y  =  <. z ,  w >. )
1513, 14sylib 196 . . . . . 6  |-  ( ( Fun  F  /\  y  e.  F )  ->  E. z E. w  y  =  <. z ,  w >. )
16 inteq 4242 . . . . . . . . . . . . . . . . 17  |-  ( y  =  <. z ,  w >.  ->  |^| y  =  |^| <.
z ,  w >. )
1716inteqd 4244 . . . . . . . . . . . . . . . 16  |-  ( y  =  <. z ,  w >.  ->  |^| |^| y  =  |^| |^|
<. z ,  w >. )
18 vex 3081 . . . . . . . . . . . . . . . . 17  |-  z  e. 
_V
19 vex 3081 . . . . . . . . . . . . . . . . 17  |-  w  e. 
_V
2018, 19op1stb 4673 . . . . . . . . . . . . . . . 16  |-  |^| |^| <. z ,  w >.  =  z
2117, 20syl6eq 2511 . . . . . . . . . . . . . . 15  |-  ( y  =  <. z ,  w >.  ->  |^| |^| y  =  z )
22 eqeq1 2458 . . . . . . . . . . . . . . 15  |-  ( x  =  |^| |^| y  ->  ( x  =  z  <->  |^| |^| y  =  z ) )
2321, 22syl5ibr 221 . . . . . . . . . . . . . 14  |-  ( x  =  |^| |^| y  ->  ( y  =  <. z ,  w >.  ->  x  =  z ) )
24 opeq1 4170 . . . . . . . . . . . . . 14  |-  ( x  =  z  ->  <. x ,  w >.  =  <. z ,  w >. )
2523, 24syl6 33 . . . . . . . . . . . . 13  |-  ( x  =  |^| |^| y  ->  ( y  =  <. z ,  w >.  ->  <. x ,  w >.  =  <. z ,  w >. )
)
2625imp 429 . . . . . . . . . . . 12  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  -> 
<. x ,  w >.  = 
<. z ,  w >. )
27 eqeq2 2469 . . . . . . . . . . . . . 14  |-  ( <.
x ,  w >.  = 
<. z ,  w >.  -> 
( y  =  <. x ,  w >.  <->  y  =  <. z ,  w >. ) )
2827biimprcd 225 . . . . . . . . . . . . 13  |-  ( y  =  <. z ,  w >.  ->  ( <. x ,  w >.  =  <. z ,  w >.  ->  y  =  <. x ,  w >. ) )
2928adantl 466 . . . . . . . . . . . 12  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  ->  ( <. x ,  w >.  =  <. z ,  w >.  ->  y  =  <. x ,  w >. )
)
3026, 29mpd 15 . . . . . . . . . . 11  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  ->  y  =  <. x ,  w >. )
3130ancoms 453 . . . . . . . . . 10  |-  ( ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y )  -> 
y  =  <. x ,  w >. )
3231adantl 466 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  y  =  <. x ,  w >. )
3330eleq1d 2523 . . . . . . . . . . . . . . 15  |-  ( ( x  =  |^| |^| y  /\  y  =  <. z ,  w >. )  ->  ( y  e.  F  <->  <.
x ,  w >.  e.  F ) )
3433adantl 466 . . . . . . . . . . . . . 14  |-  ( ( Fun  F  /\  (
x  =  |^| |^| y  /\  y  =  <. z ,  w >. )
)  ->  ( y  e.  F  <->  <. x ,  w >.  e.  F ) )
35 funopfv 5843 . . . . . . . . . . . . . . 15  |-  ( Fun 
F  ->  ( <. x ,  w >.  e.  F  ->  ( F `  x
)  =  w ) )
3635adantr 465 . . . . . . . . . . . . . 14  |-  ( ( Fun  F  /\  (
x  =  |^| |^| y  /\  y  =  <. z ,  w >. )
)  ->  ( <. x ,  w >.  e.  F  ->  ( F `  x
)  =  w ) )
3734, 36sylbid 215 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\  (
x  =  |^| |^| y  /\  y  =  <. z ,  w >. )
)  ->  ( y  e.  F  ->  ( F `
 x )  =  w ) )
3837exp32 605 . . . . . . . . . . . 12  |-  ( Fun 
F  ->  ( x  =  |^| |^| y  ->  (
y  =  <. z ,  w >.  ->  ( y  e.  F  ->  ( F `  x )  =  w ) ) ) )
3938com24 87 . . . . . . . . . . 11  |-  ( Fun 
F  ->  ( y  e.  F  ->  ( y  =  <. z ,  w >.  ->  ( x  = 
|^| |^| y  ->  ( F `  x )  =  w ) ) ) )
4039imp43 595 . . . . . . . . . 10  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  ( F `  x )  =  w )
4140opeq2d 4177 . . . . . . . . 9  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  <. x ,  ( F `  x )
>.  =  <. x ,  w >. )
4232, 41eqtr4d 2498 . . . . . . . 8  |-  ( ( ( Fun  F  /\  y  e.  F )  /\  ( y  =  <. z ,  w >.  /\  x  =  |^| |^| y ) )  ->  y  =  <. x ,  ( F `  x ) >. )
4342exp32 605 . . . . . . 7  |-  ( ( Fun  F  /\  y  e.  F )  ->  (
y  =  <. z ,  w >.  ->  ( x  =  |^| |^| y  ->  y  =  <. x ,  ( F `  x ) >. )
) )
4443exlimdvv 1692 . . . . . 6  |-  ( ( Fun  F  /\  y  e.  F )  ->  ( E. z E. w  y  =  <. z ,  w >.  ->  ( x  = 
|^| |^| y  ->  y  =  <. x ,  ( F `  x )
>. ) ) )
4515, 44mpd 15 . . . . 5  |-  ( ( Fun  F  /\  y  e.  F )  ->  (
x  =  |^| |^| y  ->  y  =  <. x ,  ( F `  x ) >. )
)
4645adantrl 715 . . . 4  |-  ( ( Fun  F  /\  (
x  e.  dom  F  /\  y  e.  F
) )  ->  (
x  =  |^| |^| y  ->  y  =  <. x ,  ( F `  x ) >. )
)
47 inteq 4242 . . . . . 6  |-  ( y  =  <. x ,  ( F `  x )
>.  ->  |^| y  =  |^| <.
x ,  ( F `
 x ) >.
)
4847inteqd 4244 . . . . 5  |-  ( y  =  <. x ,  ( F `  x )
>.  ->  |^| |^| y  =  |^| |^|
<. x ,  ( F `
 x ) >.
)
49 vex 3081 . . . . . 6  |-  x  e. 
_V
50 fvex 5812 . . . . . 6  |-  ( F `
 x )  e. 
_V
5149, 50op1stb 4673 . . . . 5  |-  |^| |^| <. x ,  ( F `  x ) >.  =  x
5248, 51syl6req 2512 . . . 4  |-  ( y  =  <. x ,  ( F `  x )
>.  ->  x  =  |^| |^| y )
5346, 52impbid1 203 . . 3  |-  ( ( Fun  F  /\  (
x  e.  dom  F  /\  y  e.  F
) )  ->  (
x  =  |^| |^| y  <->  y  =  <. x ,  ( F `  x )
>. ) )
5453ex 434 . 2  |-  ( Fun 
F  ->  ( (
x  e.  dom  F  /\  y  e.  F
)  ->  ( x  =  |^| |^| y  <->  y  =  <. x ,  ( F `
 x ) >.
) ) )
553, 4, 6, 10, 54en3d 7459 1  |-  ( Fun 
F  ->  dom  F  ~~  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   _Vcvv 3078    C_ wss 3439   <.cop 3994   |^|cint 4239   class class class wbr 4403    X. cxp 4949   dom cdm 4951   Rel wrel 4956   Fun wfun 5523   ` cfv 5529    ~~ cen 7420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-en 7424
This theorem is referenced by:  fundmeng  7497  infmap2  8502  heicant  28597
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