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Theorem fcoinver 27674
Description: Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 27675 (Contributed by Thierry Arnoux, 3-Jan-2020.)
Assertion
Ref Expression
fcoinver  |-  ( F  Fn  X  ->  ( `' F  o.  F
)  Er  X )

Proof of Theorem fcoinver
StepHypRef Expression
1 relco 5488 . . 3  |-  Rel  ( `' F  o.  F
)
21a1i 11 . 2  |-  ( F  Fn  X  ->  Rel  ( `' F  o.  F
) )
3 dmco 5498 . . 3  |-  dom  ( `' F  o.  F
)  =  ( `' F " dom  `' F )
4 df-rn 4999 . . . . 5  |-  ran  F  =  dom  `' F
54imaeq2i 5323 . . . 4  |-  ( `' F " ran  F
)  =  ( `' F " dom  `' F )
6 cnvimarndm 5346 . . . . 5  |-  ( `' F " ran  F
)  =  dom  F
7 fndm 5662 . . . . 5  |-  ( F  Fn  X  ->  dom  F  =  X )
86, 7syl5eq 2507 . . . 4  |-  ( F  Fn  X  ->  ( `' F " ran  F
)  =  X )
95, 8syl5eqr 2509 . . 3  |-  ( F  Fn  X  ->  ( `' F " dom  `' F )  =  X )
103, 9syl5eq 2507 . 2  |-  ( F  Fn  X  ->  dom  ( `' F  o.  F
)  =  X )
11 cnvco 5177 . . . . 5  |-  `' ( `' F  o.  F
)  =  ( `' F  o.  `' `' F )
12 cnvcnvss 5445 . . . . . 6  |-  `' `' F  C_  F
13 coss2 5148 . . . . . 6  |-  ( `' `' F  C_  F  -> 
( `' F  o.  `' `' F )  C_  ( `' F  o.  F
) )
1412, 13ax-mp 5 . . . . 5  |-  ( `' F  o.  `' `' F )  C_  ( `' F  o.  F
)
1511, 14eqsstri 3519 . . . 4  |-  `' ( `' F  o.  F
)  C_  ( `' F  o.  F )
1615a1i 11 . . 3  |-  ( F  Fn  X  ->  `' ( `' F  o.  F
)  C_  ( `' F  o.  F )
)
17 coass 5509 . . . . 5  |-  ( ( `' F  o.  F
)  o.  ( `' F  o.  F ) )  =  ( `' F  o.  ( F  o.  ( `' F  o.  F ) ) )
18 coass 5509 . . . . . . 7  |-  ( ( F  o.  `' F
)  o.  F )  =  ( F  o.  ( `' F  o.  F
) )
19 fnfun 5660 . . . . . . . . . 10  |-  ( F  Fn  X  ->  Fun  F )
20 funcocnv2 5822 . . . . . . . . . 10  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
2119, 20syl 16 . . . . . . . . 9  |-  ( F  Fn  X  ->  ( F  o.  `' F
)  =  (  _I  |`  ran  F ) )
2221coeq1d 5153 . . . . . . . 8  |-  ( F  Fn  X  ->  (
( F  o.  `' F )  o.  F
)  =  ( (  _I  |`  ran  F )  o.  F ) )
23 dffn3 5720 . . . . . . . . 9  |-  ( F  Fn  X  <->  F : X
--> ran  F )
24 fcoi2 5742 . . . . . . . . 9  |-  ( F : X --> ran  F  ->  ( (  _I  |`  ran  F
)  o.  F )  =  F )
2523, 24sylbi 195 . . . . . . . 8  |-  ( F  Fn  X  ->  (
(  _I  |`  ran  F
)  o.  F )  =  F )
2622, 25eqtrd 2495 . . . . . . 7  |-  ( F  Fn  X  ->  (
( F  o.  `' F )  o.  F
)  =  F )
2718, 26syl5eqr 2509 . . . . . 6  |-  ( F  Fn  X  ->  ( F  o.  ( `' F  o.  F )
)  =  F )
2827coeq2d 5154 . . . . 5  |-  ( F  Fn  X  ->  ( `' F  o.  ( F  o.  ( `' F  o.  F )
) )  =  ( `' F  o.  F
) )
2917, 28syl5eq 2507 . . . 4  |-  ( F  Fn  X  ->  (
( `' F  o.  F )  o.  ( `' F  o.  F
) )  =  ( `' F  o.  F
) )
30 ssid 3508 . . . 4  |-  ( `' F  o.  F ) 
C_  ( `' F  o.  F )
3129, 30syl6eqss 3539 . . 3  |-  ( F  Fn  X  ->  (
( `' F  o.  F )  o.  ( `' F  o.  F
) )  C_  ( `' F  o.  F
) )
3216, 31unssd 3666 . 2  |-  ( F  Fn  X  ->  ( `' ( `' F  o.  F )  u.  (
( `' F  o.  F )  o.  ( `' F  o.  F
) ) )  C_  ( `' F  o.  F
) )
33 df-er 7303 . 2  |-  ( ( `' F  o.  F
)  Er  X  <->  ( Rel  ( `' F  o.  F
)  /\  dom  ( `' F  o.  F )  =  X  /\  ( `' ( `' F  o.  F )  u.  (
( `' F  o.  F )  o.  ( `' F  o.  F
) ) )  C_  ( `' F  o.  F
) ) )
342, 10, 32, 33syl3anbrc 1178 1  |-  ( F  Fn  X  ->  ( `' F  o.  F
)  Er  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    u. cun 3459    C_ wss 3461    _I cid 4779   `'ccnv 4987   dom cdm 4988   ran crn 4989    |` cres 4990   "cima 4991    o. ccom 4992   Rel wrel 4993   Fun wfun 5564    Fn wfn 5565   -->wf 5566    Er wer 7300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-er 7303
This theorem is referenced by:  qtophaus  28074
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