MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isoval Structured version   Unicode version

Theorem isoval 15614
Description: The isomorphisms are the domain of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 21-May-2020.)
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  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfval.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
2 isofval 15606 . . . . 5  |-  ( C  e.  Cat  ->  (  Iso  `  C )  =  ( ( z  e. 
_V  |->  dom  z )  o.  (Inv `  C )
) )
31, 2syl 17 . . . 4  |-  ( ph  ->  (  Iso  `  C
)  =  ( ( z  e.  _V  |->  dom  z )  o.  (Inv `  C ) ) )
4 isoval.n . . . 4  |-  I  =  (  Iso  `  C
)
5 invfval.n . . . . 5  |-  N  =  (Inv `  C )
65coeq2i 5006 . . . 4  |-  ( ( z  e.  _V  |->  dom  z )  o.  N
)  =  ( ( z  e.  _V  |->  dom  z )  o.  (Inv `  C ) )
73, 4, 63eqtr4g 2486 . . 3  |-  ( ph  ->  I  =  ( ( z  e.  _V  |->  dom  z )  o.  N
) )
87oveqd 6313 . 2  |-  ( ph  ->  ( X I Y )  =  ( X ( ( z  e. 
_V  |->  dom  z )  o.  N ) Y ) )
9 eqid 2420 . . . . . 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 ) ) )
10 ovex 6324 . . . . . . 7  |-  ( x (Sect `  C )
y )  e.  _V
1110inex1 4557 . . . . . 6  |-  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) )  e.  _V
129, 11fnmpt2i 6867 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) )  Fn  ( B  X.  B )
13 invfval.b . . . . . . 7  |-  B  =  ( Base `  C
)
14 invfval.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
15 invfval.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
16 eqid 2420 . . . . . . 7  |-  (Sect `  C )  =  (Sect `  C )
1713, 5, 1, 14, 15, 16invffval 15607 . . . . . 6  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) ) )
1817fneq1d 5675 . . . . 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 ) ) )
1912, 18mpbiri 236 . . . 4  |-  ( ph  ->  N  Fn  ( B  X.  B ) )
20 opelxpi 4877 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
2114, 15, 20syl2anc 665 . . . 4  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
22 fvco2 5947 . . . 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 >. ) ) )
2319, 21, 22syl2anc 665 . . 3  |-  ( ph  ->  ( ( ( z  e.  _V  |->  dom  z
)  o.  N ) `
 <. X ,  Y >. )  =  ( ( z  e.  _V  |->  dom  z ) `  ( N `  <. X ,  Y >. ) ) )
24 df-ov 6299 . . 3  |-  ( X ( ( z  e. 
_V  |->  dom  z )  o.  N ) Y )  =  ( ( ( z  e.  _V  |->  dom  z )  o.  N
) `  <. X ,  Y >. )
25 ovex 6324 . . . . 5  |-  ( X N Y )  e. 
_V
26 dmeq 5046 . . . . . 6  |-  ( z  =  ( X N Y )  ->  dom  z  =  dom  ( X N Y ) )
27 eqid 2420 . . . . . 6  |-  ( z  e.  _V  |->  dom  z
)  =  ( z  e.  _V  |->  dom  z
)
2825dmex 6731 . . . . . 6  |-  dom  ( X N Y )  e. 
_V
2926, 27, 28fvmpt 5955 . . . . 5  |-  ( ( X N Y )  e.  _V  ->  (
( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  dom  ( X N Y ) )
3025, 29ax-mp 5 . . . 4  |-  ( ( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  dom  ( X N Y )
31 df-ov 6299 . . . . 5  |-  ( X N Y )  =  ( N `  <. X ,  Y >. )
3231fveq2i 5875 . . . 4  |-  ( ( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  ( ( z  e.  _V  |->  dom  z
) `  ( N `  <. X ,  Y >. ) )
3330, 32eqtr3i 2451 . . 3  |-  dom  ( X N Y )  =  ( ( z  e. 
_V  |->  dom  z ) `  ( N `  <. X ,  Y >. )
)
3423, 24, 333eqtr4g 2486 . 2  |-  ( ph  ->  ( X ( ( z  e.  _V  |->  dom  z )  o.  N
) Y )  =  dom  ( X N Y ) )
358, 34eqtrd 2461 1  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   _Vcvv 3078    i^i cin 3432   <.cop 3999    |-> cmpt 4475    X. cxp 4843   `'ccnv 4844   dom cdm 4845    o. ccom 4849    Fn wfn 5587   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   Basecbs 15073   Catccat 15514  Sectcsect 15593  Invcinv 15594    Iso ciso 15595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-inv 15597  df-iso 15598
This theorem is referenced by:  inviso1  15615  invf  15617  invco  15620  dfiso2  15621  isohom  15625  oppciso  15630  cicsym  15653  funciso  15723  ffthiso  15778  fuciso  15824  setciso  15930  catciso  15946  rngciso  38755  rngcisoALTV  38767  ringciso  38806  ringcisoALTV  38830
  Copyright terms: Public domain W3C validator