Theorem fcoinver 29418

Description: Build an equivalence relation from a function. Two values are equivalent if they have the same image by the function. See also fcoinvbr 29419. (Contributed by Thierry Arnoux, 3-Jan-2020.)

Assertion

Ref	Expression
fcoinver	|- ( F Fn X -> ( `' F o. F ) Er X )

Proof of Theorem fcoinver

Step	Hyp	Ref	Expression
1		relco 5633	. . 3 |- Rel ( `' F o. F )
2	1	a1i 11	. 2 |- ( F Fn X -> Rel ( `' F o. F ) )
3		dmco 5643	. . 3 |- dom ( `' F o. F ) = ( `' F " dom `' F )
4		df-rn 5125	. . . . 5 |- ran F = dom `' F
5	4	imaeq2i 5464	. . . 4 |- ( `' F " ran F ) = ( `' F " dom `' F )
6		cnvimarndm 5486	. . . . 5 |- ( `' F " ran F ) = dom F
7		fndm 5990	. . . . 5 |- ( F Fn X -> dom F = X )
8	6, 7	syl5eq 2668	. . . 4 |- ( F Fn X -> ( `' F " ran F ) = X )
9	5, 8	syl5eqr 2670	. . 3 |- ( F Fn X -> ( `' F " dom `' F ) = X )
10	3, 9	syl5eq 2668	. 2 |- ( F Fn X -> dom ( `' F o. F ) = X )
11		cnvco 5308	. . . . 5 |- `' ( `' F o. F ) = ( `' F o. `' `' F )
12		cnvcnvss 5589	. . . . . 6 |- `' `' F C_ F
13		coss2 5278	. . . . . 6 |- ( `' `' F C_ F -> ( `' F o. `' `' F ) C_ ( `' F o. F ) )
14	12, 13	ax-mp 5	. . . . 5 |- ( `' F o. `' `' F ) C_ ( `' F o. F )
15	11, 14	eqsstri 3635	. . . 4 |- `' ( `' F o. F ) C_ ( `' F o. F )
16	15	a1i 11	. . 3 |- ( F Fn X -> `' ( `' F o. F ) C_ ( `' F o. F ) )
17		coass 5654	. . . . 5 |- ( ( `' F o. F ) o. ( `' F o. F ) ) = ( `' F o. ( F o. ( `' F o. F ) ) )
18		coass 5654	. . . . . . 7 |- ( ( F o. `' F ) o. F ) = ( F o. ( `' F o. F ) )
19		fnfun 5988	. . . . . . . . . 10 |- ( F Fn X -> Fun F )
20		funcocnv2 6161	. . . . . . . . . 10 |- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) )
21	19, 20	syl 17	. . . . . . . . 9 |- ( F Fn X -> ( F o. `' F ) = ( _I |` ran F ) )
22	21	coeq1d 5283	. . . . . . . 8 |- ( F Fn X -> ( ( F o. `' F ) o. F ) = ( ( _I |` ran F ) o. F ) )
23		dffn3 6054	. . . . . . . . 9 |- ( F Fn X <-> F : X --> ran F )
24		fcoi2 6079	. . . . . . . . 9 |- ( F : X --> ran F -> ( ( _I |` ran F ) o. F ) = F )
25	23, 24	sylbi 207	. . . . . . . 8 |- ( F Fn X -> ( ( _I |` ran F ) o. F ) = F )
26	22, 25	eqtrd 2656	. . . . . . 7 |- ( F Fn X -> ( ( F o. `' F ) o. F ) = F )
27	18, 26	syl5eqr 2670	. . . . . 6 |- ( F Fn X -> ( F o. ( `' F o. F ) ) = F )
28	27	coeq2d 5284	. . . . 5 |- ( F Fn X -> ( `' F o. ( F o. ( `' F o. F ) ) ) = ( `' F o. F ) )
29	17, 28	syl5eq 2668	. . . 4 |- ( F Fn X -> ( ( `' F o. F ) o. ( `' F o. F ) ) = ( `' F o. F ) )
30		ssid 3624	. . . 4 |- ( `' F o. F ) C_ ( `' F o. F )
31	29, 30	syl6eqss 3655	. . 3 |- ( F Fn X -> ( ( `' F o. F ) o. ( `' F o. F ) ) C_ ( `' F o. F ) )
32	16, 31	unssd 3789	. 2 |- ( F Fn X -> ( `' ( `' F o. F ) u. ( ( `' F o. F ) o. ( `' F o. F ) ) ) C_ ( `' F o. F ) )
33		df-er 7742	. 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 ) ) )
34	2, 10, 32, 33	syl3anbrc 1246	1 |- ( F Fn X -> ( `' F o. F ) Er X )

Syntax hints:   -> wi 4   = wceq 1483   u. cun 3572   C_ wss 3574   _I cid 5023   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Rel wrel 5119   Fun wfun 5882   Fn wfn 5883   -->wf 5884   Er wer 7739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-er 7742

This theorem is referenced by:  qtophaus  29903