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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.
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 2402 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
71, 2, 3, 4, 5, 6invss 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 ) ) )
107, 8, 9mpisyl 19 . 2  |-  ( ph  ->  Rel  ( X N Y ) )
11 eqid 2402 . . . . . 6  |-  (Sect `  C )  =  (Sect `  C )
123adantr 463 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  C  e.  Cat )
135adantr 463 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  Y  e.  B
)
144adantr 463 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  X  e.  B
)
151, 2, 3, 4, 5, 11isinv 15371 . . . . . . . 8  |-  ( ph  ->  ( f ( X N Y ) g  <-> 
( f ( X (Sect `  C ) Y ) g  /\  g ( Y (Sect `  C ) X ) f ) ) )
1615simplbda 622 . . . . . . 7  |-  ( (
ph  /\  f ( X N Y ) g )  ->  g ( Y (Sect `  C ) X ) f )
1716adantrr 715 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  g ( Y (Sect `  C ) X ) f )
181, 2, 3, 4, 5, 11isinv 15371 . . . . . . . 8  |-  ( ph  ->  ( f ( X N Y ) h  <-> 
( f ( X (Sect `  C ) Y ) h  /\  h ( Y (Sect `  C ) X ) f ) ) )
1918simprbda 621 . . . . . . 7  |-  ( (
ph  /\  f ( X N Y ) h )  ->  f ( X (Sect `  C ) Y ) h )
2019adantrl 714 . . . . . 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 15366 . . . . 5  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  g  =  h )
2221ex 432 . . . 4  |-  ( ph  ->  ( ( f ( X N Y ) g  /\  f ( X N Y ) h )  ->  g  =  h ) )
2322alrimiv 1740 . . 3  |-  ( ph  ->  A. h ( ( f ( X N Y ) g  /\  f ( X N Y ) h )  ->  g  =  h ) )
2423alrimivv 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 ) ) )
2610, 24, 25sylanbrc 662 1  |-  ( ph  ->  Fun  ( X N Y ) )
Colors of variables: wff setvar class
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
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