Theorem invfun 15681

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 2453	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
7	1, 2, 3, 4, 5, 6	invss 15678	. . 3 |- ( ph -> ( X N Y ) C_ ( ( X ( Hom ` C ) Y ) X. ( Y ( Hom ` C ) X ) ) )
8		relxp 4945	. . 3 |- Rel ( ( X ( Hom ` C ) Y ) X. ( Y ( Hom ` C ) X ) )
9		relss 4925	. . 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 21	. 2 |- ( ph -> Rel ( X N Y ) )
11		eqid 2453	. . . . . 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 15677	. . . . . . . 8 |- ( ph -> ( f ( X N Y ) g <-> ( f ( X (Sect ` C ) Y ) g /\ g ( Y (Sect ` C ) X ) f ) ) )
16	15	simplbda 630	. . . . . . 7 |- ( ( ph /\ f ( X N Y ) g ) -> g ( Y (Sect ` C ) X ) f )
17	16	adantrr 724	. . . . . 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 15677	. . . . . . . 8 |- ( ph -> ( f ( X N Y ) h <-> ( f ( X (Sect ` C ) Y ) h /\ h ( Y (Sect ` C ) X ) f ) ) )
19	18	simprbda 629	. . . . . . 7 |- ( ( ph /\ f ( X N Y ) h ) -> f ( X (Sect ` C ) Y ) h )
20	19	adantrl 723	. . . . . 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 15672	. . . . 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 1775	. . 3 |- ( ph -> A. h ( ( f ( X N Y ) g /\ f ( X N Y ) h ) -> g = h ) )
24	23	alrimivv 1776	. 2 |- ( ph -> A. f A. g A. h ( ( f ( X N Y ) g /\ f ( X N Y ) h ) -> g = h ) )
25		dffun2 5595	. 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 671	1 |- ( ph -> Fun ( X N Y ) )

Syntax hints:   -> wi 4   /\ wa 371   A.wal 1444   = wceq 1446   e. wcel 1889   C_ wss 3406   class class class wbr 4405   X. cxp 4835   Rel wrel 4842   Fun wfun 5579   ` cfv 5585  (class class class)co 6295   Basecbs 15133   Hom chom 15213   Catccat 15582  Sectcsect 15661  Invcinv 15662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-cat 15586  df-cid 15587  df-sect 15664  df-inv 15665

This theorem is referenced by:  inviso1  15683  invf  15685  invco  15688  idinv  15706  funciso  15791  ffthiso  15846  fuciso  15892  setciso  15998  catciso  16014  rngciso  40088  rngcisoALTV  40100  ringciso  40139  ringcisoALTV  40163