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Theorem fcoinver 28290
Description: Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 28291. (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 5340 . . 3  |-  Rel  ( `' F  o.  F
)
21a1i 11 . 2  |-  ( F  Fn  X  ->  Rel  ( `' F  o.  F
) )
3 dmco 5350 . . 3  |-  dom  ( `' F  o.  F
)  =  ( `' F " dom  `' F )
4 df-rn 4850 . . . . 5  |-  ran  F  =  dom  `' F
54imaeq2i 5172 . . . 4  |-  ( `' F " ran  F
)  =  ( `' F " dom  `' F )
6 cnvimarndm 5195 . . . . 5  |-  ( `' F " ran  F
)  =  dom  F
7 fndm 5685 . . . . 5  |-  ( F  Fn  X  ->  dom  F  =  X )
86, 7syl5eq 2517 . . . 4  |-  ( F  Fn  X  ->  ( `' F " ran  F
)  =  X )
95, 8syl5eqr 2519 . . 3  |-  ( F  Fn  X  ->  ( `' F " dom  `' F )  =  X )
103, 9syl5eq 2517 . 2  |-  ( F  Fn  X  ->  dom  ( `' F  o.  F
)  =  X )
11 cnvco 5025 . . . . 5  |-  `' ( `' F  o.  F
)  =  ( `' F  o.  `' `' F )
12 cnvcnvss 5297 . . . . . 6  |-  `' `' F  C_  F
13 coss2 4996 . . . . . 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 3448 . . . 4  |-  `' ( `' F  o.  F
)  C_  ( `' F  o.  F )
1615a1i 11 . . 3  |-  ( F  Fn  X  ->  `' ( `' F  o.  F
)  C_  ( `' F  o.  F )
)
17 coass 5361 . . . . 5  |-  ( ( `' F  o.  F
)  o.  ( `' F  o.  F ) )  =  ( `' F  o.  ( F  o.  ( `' F  o.  F ) ) )
18 coass 5361 . . . . . . 7  |-  ( ( F  o.  `' F
)  o.  F )  =  ( F  o.  ( `' F  o.  F
) )
19 fnfun 5683 . . . . . . . . . 10  |-  ( F  Fn  X  ->  Fun  F )
20 funcocnv2 5852 . . . . . . . . . 10  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
2119, 20syl 17 . . . . . . . . 9  |-  ( F  Fn  X  ->  ( F  o.  `' F
)  =  (  _I  |`  ran  F ) )
2221coeq1d 5001 . . . . . . . 8  |-  ( F  Fn  X  ->  (
( F  o.  `' F )  o.  F
)  =  ( (  _I  |`  ran  F )  o.  F ) )
23 dffn3 5748 . . . . . . . . 9  |-  ( F  Fn  X  <->  F : X
--> ran  F )
24 fcoi2 5770 . . . . . . . . 9  |-  ( F : X --> ran  F  ->  ( (  _I  |`  ran  F
)  o.  F )  =  F )
2523, 24sylbi 200 . . . . . . . 8  |-  ( F  Fn  X  ->  (
(  _I  |`  ran  F
)  o.  F )  =  F )
2622, 25eqtrd 2505 . . . . . . 7  |-  ( F  Fn  X  ->  (
( F  o.  `' F )  o.  F
)  =  F )
2718, 26syl5eqr 2519 . . . . . 6  |-  ( F  Fn  X  ->  ( F  o.  ( `' F  o.  F )
)  =  F )
2827coeq2d 5002 . . . . 5  |-  ( F  Fn  X  ->  ( `' F  o.  ( F  o.  ( `' F  o.  F )
) )  =  ( `' F  o.  F
) )
2917, 28syl5eq 2517 . . . 4  |-  ( F  Fn  X  ->  (
( `' F  o.  F )  o.  ( `' F  o.  F
) )  =  ( `' F  o.  F
) )
30 ssid 3437 . . . 4  |-  ( `' F  o.  F ) 
C_  ( `' F  o.  F )
3129, 30syl6eqss 3468 . . 3  |-  ( F  Fn  X  ->  (
( `' F  o.  F )  o.  ( `' F  o.  F
) )  C_  ( `' F  o.  F
) )
3216, 31unssd 3601 . 2  |-  ( F  Fn  X  ->  ( `' ( `' F  o.  F )  u.  (
( `' F  o.  F )  o.  ( `' F  o.  F
) ) )  C_  ( `' F  o.  F
) )
33 df-er 7381 . 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 1214 1  |-  ( F  Fn  X  ->  ( `' F  o.  F
)  Er  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    u. cun 3388    C_ wss 3390    _I cid 4749   `'ccnv 4838   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842    o. ccom 4843   Rel wrel 4844   Fun wfun 5583    Fn wfn 5584   -->wf 5585    Er wer 7378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-fun 5591  df-fn 5592  df-f 5593  df-er 7381
This theorem is referenced by:  qtophaus  28737
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