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Theorem invfun 13682
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 2296 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
71, 2, 3, 4, 5, 6invss 13679 . . 3  |-  ( ph  ->  ( X N Y )  C_  ( ( X (  Hom  `  C
) Y )  X.  ( Y (  Hom  `  C ) X ) ) )
8 relxp 4810 . . 3  |-  Rel  (
( X (  Hom  `  C ) Y )  X.  ( Y (  Hom  `  C ) X ) )
9 relss 4791 . . 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 1366 . 2  |-  ( ph  ->  Rel  ( X N Y ) )
11 eqid 2296 . . . . . 6  |-  (Sect `  C )  =  (Sect `  C )
123adantr 451 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  C  e.  Cat )
135adantr 451 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  Y  e.  B
)
144adantr 451 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  X  e.  B
)
151, 2, 3, 4, 5, 11isinv 13678 . . . . . . . 8  |-  ( ph  ->  ( f ( X N Y ) g  <-> 
( f ( X (Sect `  C ) Y ) g  /\  g ( Y (Sect `  C ) X ) f ) ) )
1615simplbda 607 . . . . . . 7  |-  ( (
ph  /\  f ( X N Y ) g )  ->  g ( Y (Sect `  C ) X ) f )
1716adantrr 697 . . . . . 6  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  g ( Y (Sect `  C ) X ) f )
181, 2, 3, 4, 5, 11isinv 13678 . . . . . . . 8  |-  ( ph  ->  ( f ( X N Y ) h  <-> 
( f ( X (Sect `  C ) Y ) h  /\  h ( Y (Sect `  C ) X ) f ) ) )
1918simprbda 606 . . . . . . 7  |-  ( (
ph  /\  f ( X N Y ) h )  ->  f ( X (Sect `  C ) Y ) h )
2019adantrl 696 . . . . . 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 13674 . . . . 5  |-  ( (
ph  /\  ( f
( X N Y ) g  /\  f
( X N Y ) h ) )  ->  g  =  h )
2221ex 423 . . . 4  |-  ( ph  ->  ( ( f ( X N Y ) g  /\  f ( X N Y ) h )  ->  g  =  h ) )
2322alrimiv 1621 . . 3  |-  ( ph  ->  A. h ( ( f ( X N Y ) g  /\  f ( X N Y ) h )  ->  g  =  h ) )
2423alrimivv 1622 . 2  |-  ( ph  ->  A. f A. g A. h ( ( f ( X N Y ) g  /\  f
( X N Y ) h )  -> 
g  =  h ) )
25 dffun2 5281 . 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 645 1  |-  ( ph  ->  Fun  ( X N Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039    X. cxp 4703   Rel wrel 4710   Fun wfun 5265   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235   Catccat 13582  Sectcsect 13663  Invcinv 13664
This theorem is referenced by:  inviso1  13684  invf  13686  invco  13689  funciso  13764  ffthiso  13819  fuciso  13865  setciso  13939  catciso  13955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-cat 13586  df-cid 13587  df-sect 13666  df-inv 13667
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