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Theorem invfun 13917
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 2388 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
71, 2, 3, 4, 5, 6invss 13914 . . 3  |-  ( ph  ->  ( X N Y )  C_  ( ( X (  Hom  `  C
) Y )  X.  ( Y (  Hom  `  C ) X ) ) )
8 relxp 4924 . . 3  |-  Rel  (
( X (  Hom  `  C ) Y )  X.  ( Y (  Hom  `  C ) X ) )
9 relss 4904 . . 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, 9ee10 1382 . 2  |-  ( ph  ->  Rel  ( X N Y ) )
11 eqid 2388 . . . . . 6  |-  (Sect `  C )  =  (Sect `  C )
123adantr 452 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  C  e.  Cat )
135adantr 452 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  Y  e.  B
)
144adantr 452 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  X  e.  B
)
151, 2, 3, 4, 5, 11isinv 13913 . . . . . . . 8  |-  ( ph  ->  ( f ( X N Y ) g  <-> 
( f ( X (Sect `  C ) Y ) g  /\  g ( Y (Sect `  C ) X ) f ) ) )
1615simplbda 608 . . . . . . 7  |-  ( (
ph  /\  f ( X N Y ) g )  ->  g ( Y (Sect `  C ) X ) f )
1716adantrr 698 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  g ( Y (Sect `  C ) X ) f )
181, 2, 3, 4, 5, 11isinv 13913 . . . . . . . 8  |-  ( ph  ->  ( f ( X N Y ) h  <-> 
( f ( X (Sect `  C ) Y ) h  /\  h ( Y (Sect `  C ) X ) f ) ) )
1918simprbda 607 . . . . . . 7  |-  ( (
ph  /\  f ( X N Y ) h )  ->  f ( X (Sect `  C ) Y ) h )
2019adantrl 697 . . . . . 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 13909 . . . . 5  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  g  =  h )
2221ex 424 . . . 4  |-  ( ph  ->  ( ( f ( X N Y ) g  /\  f ( X N Y ) h )  ->  g  =  h ) )
2322alrimiv 1638 . . 3  |-  ( ph  ->  A. h ( ( f ( X N Y ) g  /\  f ( X N Y ) h )  ->  g  =  h ) )
2423alrimivv 1639 . 2  |-  ( ph  ->  A. f A. g A. h ( ( f ( X N Y ) g  /\  f
( X N Y ) h )  -> 
g  =  h ) )
25 dffun2 5405 . 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 646 1  |-  ( ph  ->  Fun  ( X N Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1717    C_ wss 3264   class class class wbr 4154    X. cxp 4817   Rel wrel 4824   Fun wfun 5389   ` cfv 5395  (class class class)co 6021   Basecbs 13397    Hom chom 13468   Catccat 13817  Sectcsect 13898  Invcinv 13899
This theorem is referenced by:  inviso1  13919  invf  13921  invco  13924  funciso  13999  ffthiso  14054  fuciso  14100  setciso  14174  catciso  14190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-cat 13821  df-cid 13822  df-sect 13901  df-inv 13902
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