Theorem invfun 15662

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 2423	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
7	1, 2, 3, 4, 5, 6	invss 15659	. . 3 |- ( ph -> ( X N Y ) C_ ( ( X ( Hom ` C ) Y ) X. ( Y ( Hom ` C ) X ) ) )
8		relxp 4959	. . 3 |- Rel ( ( X ( Hom ` C ) Y ) X. ( Y ( Hom ` C ) X ) )
9		relss 4939	. . 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 22	. 2 |- ( ph -> Rel ( X N Y ) )
11		eqid 2423	. . . . . 6 |- (Sect ` C ) = (Sect ` C )
12	3	adantr 467	. . . . . 6 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> C e. Cat )
13	5	adantr 467	. . . . . 6 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> Y e. B )
14	4	adantr 467	. . . . . 6 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> X e. B )
15	1, 2, 3, 4, 5, 11	isinv 15658	. . . . . . . 8 |- ( ph -> ( f ( X N Y ) g <-> ( f ( X (Sect ` C ) Y ) g /\ g ( Y (Sect ` C ) X ) f ) ) )
16	15	simplbda 629	. . . . . . 7 |- ( ( ph /\ f ( X N Y ) g ) -> g ( Y (Sect ` C ) X ) f )
17	16	adantrr 722	. . . . . 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 15658	. . . . . . . 8 |- ( ph -> ( f ( X N Y ) h <-> ( f ( X (Sect ` C ) Y ) h /\ h ( Y (Sect ` C ) X ) f ) ) )
19	18	simprbda 628	. . . . . . 7 |- ( ( ph /\ f ( X N Y ) h ) -> f ( X (Sect ` C ) Y ) h )
20	19	adantrl 721	. . . . . 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 15653	. . . . 5 |- ( ( ph /\ ( f ( X N Y ) g /\ f ( X N Y ) h ) ) -> g = h )
22	21	ex 436	. . . 4 |- ( ph -> ( ( f ( X N Y ) g /\ f ( X N Y ) h ) -> g = h ) )
23	22	alrimiv 1764	. . 3 |- ( ph -> A. h ( ( f ( X N Y ) g /\ f ( X N Y ) h ) -> g = h ) )
24	23	alrimivv 1765	. 2 |- ( ph -> A. f A. g A. h ( ( f ( X N Y ) g /\ f ( X N Y ) h ) -> g = h ) )
25		dffun2 5609	. 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 669	1 |- ( ph -> Fun ( X N Y ) )

Syntax hints:   -> wi 4   /\ wa 371   A.wal 1436   = wceq 1438   e. wcel 1869   C_ wss 3437   class class class wbr 4421   X. cxp 4849   Rel wrel 4856   Fun wfun 5593   ` cfv 5599  (class class class)co 6303   Basecbs 15114   Hom chom 15194   Catccat 15563  Sectcsect 15642  Invcinv 15643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-cat 15567  df-cid 15568  df-sect 15645  df-inv 15646

This theorem is referenced by:  inviso1  15664  invf  15666  invco  15669  idinv  15687  funciso  15772  ffthiso  15827  fuciso  15873  setciso  15979  catciso  15995  rngciso  39326  rngcisoALTV  39338  ringciso  39377  ringcisoALTV  39401