Theorem invfun 15010

Description: The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
invfval.b	|- B = ( Base ` C )
invfval.n	|- N = (Inv ` C )
invfval.c	|- ( ph -> C e. Cat )
invfval.x	|- ( ph -> X e. B )
invfval.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
invfun	|- ( ph -> Fun ( X N Y ) )

Proof of Theorem invfun

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		invfval.b	. . . 4 |- B = ( Base ` C )
2		invfval.n	. . . 4 |- N = (Inv ` C )
3		invfval.c	. . . 4 |- ( ph -> C e. Cat )
4		invfval.x	. . . 4 |- ( ph -> X e. B )
5		invfval.y	. . . 4 |- ( ph -> Y e. B )
6		eqid 2462	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
7	1, 2, 3, 4, 5, 6	invss 15007	. . 3 |- ( ph -> ( X N Y ) C_ ( ( X ( Hom ` C ) Y ) X. ( Y ( Hom ` C ) X ) ) )
8		relxp 5103	. . 3 |- Rel ( ( X ( Hom ` C ) Y ) X. ( Y ( Hom ` C ) X ) )
9		relss 5083	. . 3 |- ( ( X N Y ) C_ ( ( X ( Hom ` C ) Y ) X. ( Y ( Hom ` C ) X ) ) -> ( Rel ( ( X ( Hom ` C ) Y ) X. ( Y ( Hom ` C ) X ) ) -> Rel ( X N Y ) ) )
10	7, 8, 9	mpisyl 18	. 2 |- ( ph -> Rel ( X N Y ) )
11		eqid 2462	. . . . . 6 |- (Sect ` C ) = (Sect ` C )
12	3	adantr 465	. . . . . 6 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> C e. Cat )
13	5	adantr 465	. . . . . 6 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> Y e. B )
14	4	adantr 465	. . . . . 6 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> X e. B )
15	1, 2, 3, 4, 5, 11	isinv 15006	. . . . . . . 8 |- ( ph -> ( f ( X N Y ) g <-> ( f ( X (Sect ` C ) Y ) g /\ g ( Y (Sect ` C ) X ) f ) ) )
16	15	simplbda 624	. . . . . . 7 |- ( ( ph /\ f ( X N Y ) g ) -> g ( Y (Sect ` C ) X ) f )
17	16	adantrr 716	. . . . . 6 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> g ( Y (Sect ` C ) X ) f )
18	1, 2, 3, 4, 5, 11	isinv 15006	. . . . . . . 8 |- ( ph -> ( f ( X N Y ) h <-> ( f ( X (Sect ` C ) Y ) h /\ h ( Y (Sect ` C ) X ) f ) ) )
19	18	simprbda 623	. . . . . . 7 |- ( ( ph /\ f ( X N Y ) h ) -> f ( X (Sect ` C ) Y ) h )
20	19	adantrl 715	. . . . . 6 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> f ( X (Sect ` C ) Y ) h )
21	1, 11, 12, 13, 14, 17, 20	sectcan 15002	. . . . 5 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> g = h )
22	21	ex 434	. . . 4 |- ( ph -> ( ( f ( X N Y ) g /\ f ( X N Y ) h ) -> g = h ) )
23	22	alrimiv 1690	. . 3 |- ( ph -> A. h ( ( f ( X N Y ) g /\ f ( X N Y ) h ) -> g = h ) )
24	23	alrimivv 1691	. 2 |- ( ph -> A. f A. g A. h ( ( f ( X N Y ) g /\ f ( X N Y ) h ) -> g = h ) )
25		dffun2 5591	. 2 |- ( Fun ( X N Y ) <-> ( Rel ( X N Y ) /\ A. f A. g A. h ( ( f ( X N Y ) g /\ f ( X N Y ) h ) -> g = h ) ) )
26	10, 24, 25	sylanbrc 664	1 |- ( ph -> Fun ( X N Y ) )

Syntax hints:   -> wi 4   /\ wa 369   A.wal 1372   = wceq 1374   e. wcel 1762   C_ wss 3471   class class class wbr 4442   X. cxp 4992   Rel wrel 4999   Fun wfun 5575   ` cfv 5581  (class class class)co 6277   Basecbs 14481   Hom chom 14557   Catccat 14910  Sectcsect 14991  Invcinv 14992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-cat 14914  df-cid 14915  df-sect 14994  df-inv 14995

This theorem is referenced by:  inviso1  15012  invf  15014  invco  15017  funciso  15092  ffthiso  15147  fuciso  15193  setciso  15267  catciso  15283