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Theorem fcoinver 28053
Description: Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 28054 (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 5353 . . 3  |-  Rel  ( `' F  o.  F
)
21a1i 11 . 2  |-  ( F  Fn  X  ->  Rel  ( `' F  o.  F
) )
3 dmco 5363 . . 3  |-  dom  ( `' F  o.  F
)  =  ( `' F " dom  `' F )
4 df-rn 4865 . . . . 5  |-  ran  F  =  dom  `' F
54imaeq2i 5186 . . . 4  |-  ( `' F " ran  F
)  =  ( `' F " dom  `' F )
6 cnvimarndm 5209 . . . . 5  |-  ( `' F " ran  F
)  =  dom  F
7 fndm 5693 . . . . 5  |-  ( F  Fn  X  ->  dom  F  =  X )
86, 7syl5eq 2482 . . . 4  |-  ( F  Fn  X  ->  ( `' F " ran  F
)  =  X )
95, 8syl5eqr 2484 . . 3  |-  ( F  Fn  X  ->  ( `' F " dom  `' F )  =  X )
103, 9syl5eq 2482 . 2  |-  ( F  Fn  X  ->  dom  ( `' F  o.  F
)  =  X )
11 cnvco 5040 . . . . 5  |-  `' ( `' F  o.  F
)  =  ( `' F  o.  `' `' F )
12 cnvcnvss 5310 . . . . . 6  |-  `' `' F  C_  F
13 coss2 5011 . . . . . 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 3500 . . . 4  |-  `' ( `' F  o.  F
)  C_  ( `' F  o.  F )
1615a1i 11 . . 3  |-  ( F  Fn  X  ->  `' ( `' F  o.  F
)  C_  ( `' F  o.  F )
)
17 coass 5374 . . . . 5  |-  ( ( `' F  o.  F
)  o.  ( `' F  o.  F ) )  =  ( `' F  o.  ( F  o.  ( `' F  o.  F ) ) )
18 coass 5374 . . . . . . 7  |-  ( ( F  o.  `' F
)  o.  F )  =  ( F  o.  ( `' F  o.  F
) )
19 fnfun 5691 . . . . . . . . . 10  |-  ( F  Fn  X  ->  Fun  F )
20 funcocnv2 5855 . . . . . . . . . 10  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
2119, 20syl 17 . . . . . . . . 9  |-  ( F  Fn  X  ->  ( F  o.  `' F
)  =  (  _I  |`  ran  F ) )
2221coeq1d 5016 . . . . . . . 8  |-  ( F  Fn  X  ->  (
( F  o.  `' F )  o.  F
)  =  ( (  _I  |`  ran  F )  o.  F ) )
23 dffn3 5753 . . . . . . . . 9  |-  ( F  Fn  X  <->  F : X
--> ran  F )
24 fcoi2 5775 . . . . . . . . 9  |-  ( F : X --> ran  F  ->  ( (  _I  |`  ran  F
)  o.  F )  =  F )
2523, 24sylbi 198 . . . . . . . 8  |-  ( F  Fn  X  ->  (
(  _I  |`  ran  F
)  o.  F )  =  F )
2622, 25eqtrd 2470 . . . . . . 7  |-  ( F  Fn  X  ->  (
( F  o.  `' F )  o.  F
)  =  F )
2718, 26syl5eqr 2484 . . . . . 6  |-  ( F  Fn  X  ->  ( F  o.  ( `' F  o.  F )
)  =  F )
2827coeq2d 5017 . . . . 5  |-  ( F  Fn  X  ->  ( `' F  o.  ( F  o.  ( `' F  o.  F )
) )  =  ( `' F  o.  F
) )
2917, 28syl5eq 2482 . . . 4  |-  ( F  Fn  X  ->  (
( `' F  o.  F )  o.  ( `' F  o.  F
) )  =  ( `' F  o.  F
) )
30 ssid 3489 . . . 4  |-  ( `' F  o.  F ) 
C_  ( `' F  o.  F )
3129, 30syl6eqss 3520 . . 3  |-  ( F  Fn  X  ->  (
( `' F  o.  F )  o.  ( `' F  o.  F
) )  C_  ( `' F  o.  F
) )
3216, 31unssd 3648 . 2  |-  ( F  Fn  X  ->  ( `' ( `' F  o.  F )  u.  (
( `' F  o.  F )  o.  ( `' F  o.  F
) ) )  C_  ( `' F  o.  F
) )
33 df-er 7371 . 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 1189 1  |-  ( F  Fn  X  ->  ( `' F  o.  F
)  Er  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    u. cun 3440    C_ wss 3442    _I cid 4764   `'ccnv 4853   dom cdm 4854   ran crn 4855    |` cres 4856   "cima 4857    o. ccom 4858   Rel wrel 4859   Fun wfun 5595    Fn wfn 5596   -->wf 5597    Er wer 7368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-fun 5603  df-fn 5604  df-f 5605  df-er 7371
This theorem is referenced by:  qtophaus  28502
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