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Theorem isoval 15020
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 5866 . . . . . . . 8  |-  ( c  =  C  ->  (Inv `  c )  =  (Inv
`  C ) )
4 invfval.n . . . . . . . 8  |-  N  =  (Inv `  C )
53, 4syl6eqr 2526 . . . . . . 7  |-  ( c  =  C  ->  (Inv `  c )  =  N )
65coeq2d 5165 . . . . . 6  |-  ( c  =  C  ->  (
( z  e.  _V  |->  dom  z )  o.  (Inv `  c ) )  =  ( ( z  e. 
_V  |->  dom  z )  o.  N ) )
7 df-iso 15005 . . . . . 6  |-  Iso  =  ( c  e.  Cat  |->  ( ( z  e. 
_V  |->  dom  z )  o.  (Inv `  c )
) )
8 funmpt 5624 . . . . . . 7  |-  Fun  (
z  e.  _V  |->  dom  z )
9 fvex 5876 . . . . . . . 8  |-  (Inv `  C )  e.  _V
104, 9eqeltri 2551 . . . . . . 7  |-  N  e. 
_V
11 cofunexg 6748 . . . . . . 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 5950 . . . . 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 2520 . . 3  |-  ( ph  ->  I  =  ( ( z  e.  _V  |->  dom  z )  o.  N
) )
1615oveqd 6301 . 2  |-  ( ph  ->  ( X I Y )  =  ( X ( ( z  e. 
_V  |->  dom  z )  o.  N ) Y ) )
17 eqid 2467 . . . . . 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 6309 . . . . . . 7  |-  ( x (Sect `  C )
y )  e.  _V
1918inex1 4588 . . . . . 6  |-  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) )  e.  _V
2017, 19fnmpt2i 6853 . . . . 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 2467 . . . . . . 7  |-  (Sect `  C )  =  (Sect `  C )
2521, 4, 2, 22, 23, 24invffval 15013 . . . . . 6  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) ) )
2625fneq1d 5671 . . . . 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 5031 . . . . 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 5942 . . . 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 6287 . . 3  |-  ( X ( ( z  e. 
_V  |->  dom  z )  o.  N ) Y )  =  ( ( ( z  e.  _V  |->  dom  z )  o.  N
) `  <. X ,  Y >. )
33 ovex 6309 . . . . 5  |-  ( X N Y )  e. 
_V
34 dmeq 5203 . . . . . 6  |-  ( z  =  ( X N Y )  ->  dom  z  =  dom  ( X N Y ) )
35 eqid 2467 . . . . . 6  |-  ( z  e.  _V  |->  dom  z
)  =  ( z  e.  _V  |->  dom  z
)
3633dmex 6717 . . . . . 6  |-  dom  ( X N Y )  e. 
_V
3734, 35, 36fvmpt 5950 . . . . 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 6287 . . . . 5  |-  ( X N Y )  =  ( N `  <. X ,  Y >. )
4039fveq2i 5869 . . . 4  |-  ( ( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  ( ( z  e.  _V  |->  dom  z
) `  ( N `  <. X ,  Y >. ) )
4138, 40eqtr3i 2498 . . 3  |-  dom  ( X N Y )  =  ( ( z  e. 
_V  |->  dom  z ) `  ( N `  <. X ,  Y >. )
)
4231, 32, 413eqtr4g 2533 . 2  |-  ( ph  ->  ( X ( ( z  e.  _V  |->  dom  z )  o.  N
) Y )  =  dom  ( X N Y ) )
4316, 42eqtrd 2508 1  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113    i^i cin 3475   <.cop 4033    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   dom cdm 4999    o. ccom 5003   Fun wfun 5582    Fn wfn 5583   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   Basecbs 14490   Catccat 14919  Sectcsect 15000  Invcinv 15001    Iso ciso 15002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-inv 15004  df-iso 15005
This theorem is referenced by:  inviso1  15021  invf  15023  invco  15026  isohom  15027  oppciso  15032  funciso  15101  ffthiso  15156  fuciso  15202  setciso  15276  catciso  15292
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