Theorem invfun 15375

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 2402	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
7	1, 2, 3, 4, 5, 6	invss 15372	. . 3 |- ( ph -> ( X N Y ) C_ ( ( X ( Hom ` C ) Y ) X. ( Y ( Hom ` C ) X ) ) )
8		relxp 4930	. . 3 |- Rel ( ( X ( Hom ` C ) Y ) X. ( Y ( Hom ` C ) X ) )
9		relss 4910	. . 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 19	. 2 |- ( ph -> Rel ( X N Y ) )
11		eqid 2402	. . . . . 6 |- (Sect ` C ) = (Sect ` C )
12	3	adantr 463	. . . . . 6 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> C e. Cat )
13	5	adantr 463	. . . . . 6 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> Y e. B )
14	4	adantr 463	. . . . . 6 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> X e. B )
15	1, 2, 3, 4, 5, 11	isinv 15371	. . . . . . . 8 |- ( ph -> ( f ( X N Y ) g <-> ( f ( X (Sect ` C ) Y ) g /\ g ( Y (Sect ` C ) X ) f ) ) )
16	15	simplbda 622	. . . . . . 7 |- ( ( ph /\ f ( X N Y ) g ) -> g ( Y (Sect ` C ) X ) f )
17	16	adantrr 715	. . . . . 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 15371	. . . . . . . 8 |- ( ph -> ( f ( X N Y ) h <-> ( f ( X (Sect ` C ) Y ) h /\ h ( Y (Sect ` C ) X ) f ) ) )
19	18	simprbda 621	. . . . . . 7 |- ( ( ph /\ f ( X N Y ) h ) -> f ( X (Sect ` C ) Y ) h )
20	19	adantrl 714	. . . . . 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 15366	. . . . 5 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> g = h )
22	21	ex 432	. . . 4 |- ( ph -> ( ( f ( X N Y ) g /\ f ( X N Y ) h ) -> g = h ) )
23	22	alrimiv 1740	. . 3 |- ( ph -> A. h ( ( f ( X N Y ) g /\ f ( X N Y ) h ) -> g = h ) )
24	23	alrimivv 1741	. 2 |- ( ph -> A. f A. g A. h ( ( f ( X N Y ) g /\ f ( X N Y ) h ) -> g = h ) )
25		dffun2 5578	. 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 662	1 |- ( ph -> Fun ( X N Y ) )

Syntax hints:   -> wi 4   /\ wa 367   A.wal 1403   = wceq 1405   e. wcel 1842   C_ wss 3413   class class class wbr 4394   X. cxp 4820   Rel wrel 4827   Fun wfun 5562   ` cfv 5568  (class class class)co 6277   Basecbs 14839   Hom chom 14918   Catccat 15276  Sectcsect 15355  Invcinv 15356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-cat 15280  df-cid 15281  df-sect 15358  df-inv 15359

This theorem is referenced by:  inviso1  15377  invf  15379  invco  15382  idinv  15400  funciso  15485  ffthiso  15540  fuciso  15586  setciso  15692  catciso  15708  rngciso  38282  rngcisoALTV  38294  ringciso  38333  ringcisoALTV  38357