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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.
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 2453 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
71, 2, 3, 4, 5, 6invss 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 ) ) )
107, 8, 9mpisyl 21 . 2  |-  ( ph  ->  Rel  ( X N Y ) )
11 eqid 2453 . . . . . 6  |-  (Sect `  C )  =  (Sect `  C )
123adantr 467 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  C  e.  Cat )
135adantr 467 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  Y  e.  B
)
144adantr 467 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  X  e.  B
)
151, 2, 3, 4, 5, 11isinv 15677 . . . . . . . 8  |-  ( ph  ->  ( f ( X N Y ) g  <-> 
( f ( X (Sect `  C ) Y ) g  /\  g ( Y (Sect `  C ) X ) f ) ) )
1615simplbda 630 . . . . . . 7  |-  ( (
ph  /\  f ( X N Y ) g )  ->  g ( Y (Sect `  C ) X ) f )
1716adantrr 724 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  g ( Y (Sect `  C ) X ) f )
181, 2, 3, 4, 5, 11isinv 15677 . . . . . . . 8  |-  ( ph  ->  ( f ( X N Y ) h  <-> 
( f ( X (Sect `  C ) Y ) h  /\  h ( Y (Sect `  C ) X ) f ) ) )
1918simprbda 629 . . . . . . 7  |-  ( (
ph  /\  f ( X N Y ) h )  ->  f ( X (Sect `  C ) Y ) h )
2019adantrl 723 . . . . . 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 15672 . . . . 5  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  g  =  h )
2221ex 436 . . . 4  |-  ( ph  ->  ( ( f ( X N Y ) g  /\  f ( X N Y ) h )  ->  g  =  h ) )
2322alrimiv 1775 . . 3  |-  ( ph  ->  A. h ( ( f ( X N Y ) g  /\  f ( X N Y ) h )  ->  g  =  h ) )
2423alrimivv 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 ) ) )
2610, 24, 25sylanbrc 671 1  |-  ( ph  ->  Fun  ( X N Y ) )
Colors of variables: wff setvar class
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
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