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

Theorem invf 13993
Description: The inverse relation is a function from isomorphisms to isomorphisms. (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
invf  |-  ( ph  ->  ( X N Y ) : ( X I Y ) --> ( Y I X ) )

Proof of Theorem invf
StepHypRef Expression
1 invfval.b . . . . 5  |-  B  =  ( Base `  C
)
2 invfval.n . . . . 5  |-  N  =  (Inv `  C )
3 invfval.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
4 invfval.x . . . . 5  |-  ( ph  ->  X  e.  B )
5 invfval.y . . . . 5  |-  ( ph  ->  Y  e.  B )
61, 2, 3, 4, 5invfun 13989 . . . 4  |-  ( ph  ->  Fun  ( X N Y ) )
7 funfn 5482 . . . 4  |-  ( Fun  ( X N Y )  <->  ( X N Y )  Fn  dom  ( X N Y ) )
86, 7sylib 189 . . 3  |-  ( ph  ->  ( X N Y )  Fn  dom  ( X N Y ) )
9 isoval.n . . . . 5  |-  I  =  (  Iso  `  C
)
101, 2, 3, 4, 5, 9isoval 13990 . . . 4  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
1110fneq2d 5537 . . 3  |-  ( ph  ->  ( ( X N Y )  Fn  ( X I Y )  <-> 
( X N Y )  Fn  dom  ( X N Y ) ) )
128, 11mpbird 224 . 2  |-  ( ph  ->  ( X N Y )  Fn  ( X I Y ) )
13 df-rn 4889 . . . 4  |-  ran  ( X N Y )  =  dom  `' ( X N Y )
141, 2, 3, 4, 5invsym2 13988 . . . . . 6  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
1514dmeqd 5072 . . . . 5  |-  ( ph  ->  dom  `' ( X N Y )  =  dom  ( Y N X ) )
161, 2, 3, 5, 4, 9isoval 13990 . . . . 5  |-  ( ph  ->  ( Y I X )  =  dom  ( Y N X ) )
1715, 16eqtr4d 2471 . . . 4  |-  ( ph  ->  dom  `' ( X N Y )  =  ( Y I X ) )
1813, 17syl5eq 2480 . . 3  |-  ( ph  ->  ran  ( X N Y )  =  ( Y I X ) )
19 eqimss 3400 . . 3  |-  ( ran  ( X N Y )  =  ( Y I X )  ->  ran  ( X N Y )  C_  ( Y I X ) )
2018, 19syl 16 . 2  |-  ( ph  ->  ran  ( X N Y )  C_  ( Y I X ) )
21 df-f 5458 . 2  |-  ( ( X N Y ) : ( X I Y ) --> ( Y I X )  <->  ( ( X N Y )  Fn  ( X I Y )  /\  ran  ( X N Y )  C_  ( Y I X ) ) )
2212, 20, 21sylanbrc 646 1  |-  ( ph  ->  ( X N Y ) : ( X I Y ) --> ( Y I X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    C_ wss 3320   `'ccnv 4877   dom cdm 4878   ran crn 4879   Fun wfun 5448    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   Catccat 13889  Invcinv 13971    Iso ciso 13972
This theorem is referenced by:  invf1o  13994  ffthiso  14126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-cat 13893  df-cid 13894  df-sect 13973  df-inv 13974  df-iso 13975
  Copyright terms: Public domain W3C validator