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Theorem isoval 14701
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 )
isoval.n  |-  I  =  (  Iso  `  C
)
Assertion
Ref Expression
isoval  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )

Proof of Theorem isoval
Dummy variables  x  c  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isoval.n . . . 4  |-  I  =  (  Iso  `  C
)
2 invfval.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
3 fveq2 5689 . . . . . . . 8  |-  ( c  =  C  ->  (Inv `  c )  =  (Inv
`  C ) )
4 invfval.n . . . . . . . 8  |-  N  =  (Inv `  C )
53, 4syl6eqr 2491 . . . . . . 7  |-  ( c  =  C  ->  (Inv `  c )  =  N )
65coeq2d 5000 . . . . . 6  |-  ( c  =  C  ->  (
( z  e.  _V  |->  dom  z )  o.  (Inv `  c ) )  =  ( ( z  e. 
_V  |->  dom  z )  o.  N ) )
7 df-iso 14686 . . . . . 6  |-  Iso  =  ( c  e.  Cat  |->  ( ( z  e. 
_V  |->  dom  z )  o.  (Inv `  c )
) )
8 funmpt 5452 . . . . . . 7  |-  Fun  (
z  e.  _V  |->  dom  z )
9 fvex 5699 . . . . . . . 8  |-  (Inv `  C )  e.  _V
104, 9eqeltri 2511 . . . . . . 7  |-  N  e. 
_V
11 cofunexg 6539 . . . . . . 7  |-  ( ( Fun  ( z  e. 
_V  |->  dom  z )  /\  N  e.  _V )  ->  ( ( z  e.  _V  |->  dom  z
)  o.  N )  e.  _V )
128, 10, 11mp2an 672 . . . . . 6  |-  ( ( z  e.  _V  |->  dom  z )  o.  N
)  e.  _V
136, 7, 12fvmpt 5772 . . . . 5  |-  ( C  e.  Cat  ->  (  Iso  `  C )  =  ( ( z  e. 
_V  |->  dom  z )  o.  N ) )
142, 13syl 16 . . . 4  |-  ( ph  ->  (  Iso  `  C
)  =  ( ( z  e.  _V  |->  dom  z )  o.  N
) )
151, 14syl5eq 2485 . . 3  |-  ( ph  ->  I  =  ( ( z  e.  _V  |->  dom  z )  o.  N
) )
1615oveqd 6106 . 2  |-  ( ph  ->  ( X I Y )  =  ( X ( ( z  e. 
_V  |->  dom  z )  o.  N ) Y ) )
17 eqid 2441 . . . . . 6  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) )  =  ( x  e.  B , 
y  e.  B  |->  ( ( x (Sect `  C ) y )  i^i  `' ( y (Sect `  C )
x ) ) )
18 ovex 6114 . . . . . . 7  |-  ( x (Sect `  C )
y )  e.  _V
1918inex1 4431 . . . . . 6  |-  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) )  e.  _V
2017, 19fnmpt2i 6641 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) )  Fn  ( B  X.  B )
21 invfval.b . . . . . . 7  |-  B  =  ( Base `  C
)
22 invfval.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
23 invfval.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
24 eqid 2441 . . . . . . 7  |-  (Sect `  C )  =  (Sect `  C )
2521, 4, 2, 22, 23, 24invffval 14694 . . . . . 6  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) ) )
2625fneq1d 5499 . . . . 5  |-  ( ph  ->  ( N  Fn  ( B  X.  B )  <->  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C )
y )  i^i  `' ( y (Sect `  C ) x ) ) )  Fn  ( B  X.  B ) ) )
2720, 26mpbiri 233 . . . 4  |-  ( ph  ->  N  Fn  ( B  X.  B ) )
28 opelxpi 4869 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
2922, 23, 28syl2anc 661 . . . 4  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
30 fvco2 5764 . . . 4  |-  ( ( N  Fn  ( B  X.  B )  /\  <. X ,  Y >.  e.  ( B  X.  B
) )  ->  (
( ( z  e. 
_V  |->  dom  z )  o.  N ) `  <. X ,  Y >. )  =  ( ( z  e.  _V  |->  dom  z
) `  ( N `  <. X ,  Y >. ) ) )
3127, 29, 30syl2anc 661 . . 3  |-  ( ph  ->  ( ( ( z  e.  _V  |->  dom  z
)  o.  N ) `
 <. X ,  Y >. )  =  ( ( z  e.  _V  |->  dom  z ) `  ( N `  <. X ,  Y >. ) ) )
32 df-ov 6092 . . 3  |-  ( X ( ( z  e. 
_V  |->  dom  z )  o.  N ) Y )  =  ( ( ( z  e.  _V  |->  dom  z )  o.  N
) `  <. X ,  Y >. )
33 ovex 6114 . . . . 5  |-  ( X N Y )  e. 
_V
34 dmeq 5038 . . . . . 6  |-  ( z  =  ( X N Y )  ->  dom  z  =  dom  ( X N Y ) )
35 eqid 2441 . . . . . 6  |-  ( z  e.  _V  |->  dom  z
)  =  ( z  e.  _V  |->  dom  z
)
3633dmex 6509 . . . . . 6  |-  dom  ( X N Y )  e. 
_V
3734, 35, 36fvmpt 5772 . . . . 5  |-  ( ( X N Y )  e.  _V  ->  (
( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  dom  ( X N Y ) )
3833, 37ax-mp 5 . . . 4  |-  ( ( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  dom  ( X N Y )
39 df-ov 6092 . . . . 5  |-  ( X N Y )  =  ( N `  <. X ,  Y >. )
4039fveq2i 5692 . . . 4  |-  ( ( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  ( ( z  e.  _V  |->  dom  z
) `  ( N `  <. X ,  Y >. ) )
4138, 40eqtr3i 2463 . . 3  |-  dom  ( X N Y )  =  ( ( z  e. 
_V  |->  dom  z ) `  ( N `  <. X ,  Y >. )
)
4231, 32, 413eqtr4g 2498 . 2  |-  ( ph  ->  ( X ( ( z  e.  _V  |->  dom  z )  o.  N
) Y )  =  dom  ( X N Y ) )
4316, 42eqtrd 2473 1  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2970    i^i cin 3325   <.cop 3881    e. cmpt 4348    X. cxp 4836   `'ccnv 4837   dom cdm 4838    o. ccom 4842   Fun wfun 5410    Fn wfn 5411   ` cfv 5416  (class class class)co 6089    e. cmpt2 6091   Basecbs 14172   Catccat 14600  Sectcsect 14681  Invcinv 14682    Iso ciso 14683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-inv 14685  df-iso 14686
This theorem is referenced by:  inviso1  14702  invf  14704  invco  14707  isohom  14708  oppciso  14713  funciso  14782  ffthiso  14837  fuciso  14883  setciso  14957  catciso  14973
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