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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.
StepHypRef 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 )
71, 2, 3, 4, 5, 6invss 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 ) ) )
107, 8, 9mpisyl 18 . 2  |-  ( ph  ->  Rel  ( X N Y ) )
11 eqid 2462 . . . . . 6  |-  (Sect `  C )  =  (Sect `  C )
123adantr 465 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  C  e.  Cat )
135adantr 465 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  Y  e.  B
)
144adantr 465 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  X  e.  B
)
151, 2, 3, 4, 5, 11isinv 15006 . . . . . . . 8  |-  ( ph  ->  ( f ( X N Y ) g  <-> 
( f ( X (Sect `  C ) Y ) g  /\  g ( Y (Sect `  C ) X ) f ) ) )
1615simplbda 624 . . . . . . 7  |-  ( (
ph  /\  f ( X N Y ) g )  ->  g ( Y (Sect `  C ) X ) f )
1716adantrr 716 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  g ( Y (Sect `  C ) X ) f )
181, 2, 3, 4, 5, 11isinv 15006 . . . . . . . 8  |-  ( ph  ->  ( f ( X N Y ) h  <-> 
( f ( X (Sect `  C ) Y ) h  /\  h ( Y (Sect `  C ) X ) f ) ) )
1918simprbda 623 . . . . . . 7  |-  ( (
ph  /\  f ( X N Y ) h )  ->  f ( X (Sect `  C ) Y ) h )
2019adantrl 715 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  f ( X (Sect `  C ) Y ) h )
211, 11, 12, 13, 14, 17, 20sectcan 15002 . . . . 5  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  g  =  h )
2221ex 434 . . . 4  |-  ( ph  ->  ( ( f ( X N Y ) g  /\  f ( X N Y ) h )  ->  g  =  h ) )
2322alrimiv 1690 . . 3  |-  ( ph  ->  A. h ( ( f ( X N Y ) g  /\  f ( X N Y ) h )  ->  g  =  h ) )
2423alrimivv 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 ) ) )
2610, 24, 25sylanbrc 664 1  |-  ( ph  ->  Fun  ( X N Y ) )
Colors of variables: wff setvar class
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
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