Theorem setcinv 16740

Description: An inverse in the category of sets is the converse operation. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
setcmon.c	|- C = ( SetCat ` U )
setcmon.u	|- ( ph -> U e. V )
setcmon.x	|- ( ph -> X e. U )
setcmon.y	|- ( ph -> Y e. U )
setcinv.n	|- N = (Inv ` C )

Assertion

Ref	Expression
setcinv	|- ( ph -> ( F ( X N Y ) G <-> ( F : X -1-1-onto-> Y /\ G = `' F ) ) )

Proof of Theorem setcinv

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( Base ` C ) = ( Base ` C )
2		setcinv.n	. . 3 |- N = (Inv ` C )
3		setcmon.u	. . . 4 |- ( ph -> U e. V )
4		setcmon.c	. . . . 5 |- C = ( SetCat ` U )
5	4	setccat 16735	. . . 4 |- ( U e. V -> C e. Cat )
6	3, 5	syl 17	. . 3 |- ( ph -> C e. Cat )
7		setcmon.x	. . . 4 |- ( ph -> X e. U )
8	4, 3	setcbas 16728	. . . 4 |- ( ph -> U = ( Base ` C ) )
9	7, 8	eleqtrd 2703	. . 3 |- ( ph -> X e. ( Base ` C ) )
10		setcmon.y	. . . 4 |- ( ph -> Y e. U )
11	10, 8	eleqtrd 2703	. . 3 |- ( ph -> Y e. ( Base ` C ) )
12		eqid 2622	. . 3 |- (Sect ` C ) = (Sect ` C )
13	1, 2, 6, 9, 11, 12	isinv 16420	. 2 |- ( ph -> ( F ( X N Y ) G <-> ( F ( X (Sect ` C ) Y ) G /\ G ( Y (Sect ` C ) X ) F ) ) )
14	4, 3, 7, 10, 12	setcsect 16739	. . . . 5 |- ( ph -> ( F ( X (Sect ` C ) Y ) G <-> ( F : X --> Y /\ G : Y --> X /\ ( G o. F ) = ( _I |` X ) ) ) )
15		df-3an 1039	. . . . 5 |- ( ( F : X --> Y /\ G : Y --> X /\ ( G o. F ) = ( _I |` X ) ) <-> ( ( F : X --> Y /\ G : Y --> X ) /\ ( G o. F ) = ( _I |` X ) ) )
16	14, 15	syl6bb 276	. . . 4 |- ( ph -> ( F ( X (Sect ` C ) Y ) G <-> ( ( F : X --> Y /\ G : Y --> X ) /\ ( G o. F ) = ( _I |` X ) ) ) )
17	4, 3, 10, 7, 12	setcsect 16739	. . . . 5 |- ( ph -> ( G ( Y (Sect ` C ) X ) F <-> ( G : Y --> X /\ F : X --> Y /\ ( F o. G ) = ( _I |` Y ) ) ) )
18		3ancoma 1045	. . . . . 6 |- ( ( G : Y --> X /\ F : X --> Y /\ ( F o. G ) = ( _I |` Y ) ) <-> ( F : X --> Y /\ G : Y --> X /\ ( F o. G ) = ( _I |` Y ) ) )
19		df-3an 1039	. . . . . 6 |- ( ( F : X --> Y /\ G : Y --> X /\ ( F o. G ) = ( _I |` Y ) ) <-> ( ( F : X --> Y /\ G : Y --> X ) /\ ( F o. G ) = ( _I |` Y ) ) )
20	18, 19	bitri 264	. . . . 5 |- ( ( G : Y --> X /\ F : X --> Y /\ ( F o. G ) = ( _I |` Y ) ) <-> ( ( F : X --> Y /\ G : Y --> X ) /\ ( F o. G ) = ( _I |` Y ) ) )
21	17, 20	syl6bb 276	. . . 4 |- ( ph -> ( G ( Y (Sect ` C ) X ) F <-> ( ( F : X --> Y /\ G : Y --> X ) /\ ( F o. G ) = ( _I |` Y ) ) ) )
22	16, 21	anbi12d 747	. . 3 |- ( ph -> ( ( F ( X (Sect ` C ) Y ) G /\ G ( Y (Sect ` C ) X ) F ) <-> ( ( ( F : X --> Y /\ G : Y --> X ) /\ ( G o. F ) = ( _I |` X ) ) /\ ( ( F : X --> Y /\ G : Y --> X ) /\ ( F o. G ) = ( _I |` Y ) ) ) ) )
23		anandi 871	. . 3 |- ( ( ( F : X --> Y /\ G : Y --> X ) /\ ( ( G o. F ) = ( _I |` X ) /\ ( F o. G ) = ( _I |` Y ) ) ) <-> ( ( ( F : X --> Y /\ G : Y --> X ) /\ ( G o. F ) = ( _I |` X ) ) /\ ( ( F : X --> Y /\ G : Y --> X ) /\ ( F o. G ) = ( _I |` Y ) ) ) )
24	22, 23	syl6bbr 278	. 2 |- ( ph -> ( ( F ( X (Sect ` C ) Y ) G /\ G ( Y (Sect ` C ) X ) F ) <-> ( ( F : X --> Y /\ G : Y --> X ) /\ ( ( G o. F ) = ( _I |` X ) /\ ( F o. G ) = ( _I |` Y ) ) ) ) )
25		fcof1o 6551	. . . . . 6 |- ( ( ( F : X --> Y /\ G : Y --> X ) /\ ( ( F o. G ) = ( _I |` Y ) /\ ( G o. F ) = ( _I |` X ) ) ) -> ( F : X -1-1-onto-> Y /\ `' F = G ) )
26		eqcom 2629	. . . . . . 7 |- ( `' F = G <-> G = `' F )
27	26	anbi2i 730	. . . . . 6 |- ( ( F : X -1-1-onto-> Y /\ `' F = G ) <-> ( F : X -1-1-onto-> Y /\ G = `' F ) )
28	25, 27	sylib 208	. . . . 5 |- ( ( ( F : X --> Y /\ G : Y --> X ) /\ ( ( F o. G ) = ( _I |` Y ) /\ ( G o. F ) = ( _I |` X ) ) ) -> ( F : X -1-1-onto-> Y /\ G = `' F ) )
29	28	ancom2s 844	. . . 4 |- ( ( ( F : X --> Y /\ G : Y --> X ) /\ ( ( G o. F ) = ( _I |` X ) /\ ( F o. G ) = ( _I |` Y ) ) ) -> ( F : X -1-1-onto-> Y /\ G = `' F ) )
30	29	adantl 482	. . 3 |- ( ( ph /\ ( ( F : X --> Y /\ G : Y --> X ) /\ ( ( G o. F ) = ( _I |` X ) /\ ( F o. G ) = ( _I |` Y ) ) ) ) -> ( F : X -1-1-onto-> Y /\ G = `' F ) )
31		f1of 6137	. . . . 5 |- ( F : X -1-1-onto-> Y -> F : X --> Y )
32	31	ad2antrl 764	. . . 4 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> F : X --> Y )
33		f1ocnv 6149	. . . . . . 7 |- ( F : X -1-1-onto-> Y -> `' F : Y -1-1-onto-> X )
34	33	ad2antrl 764	. . . . . 6 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> `' F : Y -1-1-onto-> X )
35		f1oeq1 6127	. . . . . . 7 |- ( G = `' F -> ( G : Y -1-1-onto-> X <-> `' F : Y -1-1-onto-> X ) )
36	35	ad2antll 765	. . . . . 6 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> ( G : Y -1-1-onto-> X <-> `' F : Y -1-1-onto-> X ) )
37	34, 36	mpbird 247	. . . . 5 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> G : Y -1-1-onto-> X )
38		f1of 6137	. . . . 5 |- ( G : Y -1-1-onto-> X -> G : Y --> X )
39	37, 38	syl 17	. . . 4 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> G : Y --> X )
40		simprr 796	. . . . . . 7 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> G = `' F )
41	40	coeq1d 5283	. . . . . 6 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> ( G o. F ) = ( `' F o. F ) )
42		f1ococnv1 6165	. . . . . . 7 |- ( F : X -1-1-onto-> Y -> ( `' F o. F ) = ( _I |` X ) )
43	42	ad2antrl 764	. . . . . 6 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> ( `' F o. F ) = ( _I |` X ) )
44	41, 43	eqtrd 2656	. . . . 5 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> ( G o. F ) = ( _I |` X ) )
45	40	coeq2d 5284	. . . . . 6 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> ( F o. G ) = ( F o. `' F ) )
46		f1ococnv2 6163	. . . . . . 7 |- ( F : X -1-1-onto-> Y -> ( F o. `' F ) = ( _I |` Y ) )
47	46	ad2antrl 764	. . . . . 6 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> ( F o. `' F ) = ( _I |` Y ) )
48	45, 47	eqtrd 2656	. . . . 5 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> ( F o. G ) = ( _I |` Y ) )
49	44, 48	jca 554	. . . 4 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> ( ( G o. F ) = ( _I |` X ) /\ ( F o. G ) = ( _I |` Y ) ) )
50	32, 39, 49	jca31 557	. . 3 |- ( ( ph /\ ( F : X -1-1-onto-> Y /\ G = `' F ) ) -> ( ( F : X --> Y /\ G : Y --> X ) /\ ( ( G o. F ) = ( _I |` X ) /\ ( F o. G ) = ( _I |` Y ) ) ) )
51	30, 50	impbida 877	. 2 |- ( ph -> ( ( ( F : X --> Y /\ G : Y --> X ) /\ ( ( G o. F ) = ( _I |` X ) /\ ( F o. G ) = ( _I |` Y ) ) ) <-> ( F : X -1-1-onto-> Y /\ G = `' F ) ) )
52	13, 24, 51	3bitrd 294	1 |- ( ph -> ( F ( X N Y ) G <-> ( F : X -1-1-onto-> Y /\ G = `' F ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Catccat 16325  Sectcsect 16404  Invcinv 16405   SetCatcsetc 16725

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-sect 16407  df-inv 16408  df-setc 16726

This theorem is referenced by:  setciso  16741  yonedainv  16921