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

Theorem invf 15259
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 15255 . . . 4  |-  ( ph  ->  Fun  ( X N Y ) )
7 funfn 5599 . . . 4  |-  ( Fun  ( X N Y )  <->  ( X N Y )  Fn  dom  ( X N Y ) )
86, 7sylib 196 . . 3  |-  ( ph  ->  ( X N Y )  Fn  dom  ( X N Y ) )
9 isoval.n . . . . 5  |-  I  =  (  Iso  `  C
)
101, 2, 3, 4, 5, 9isoval 15256 . . . 4  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
1110fneq2d 5654 . . 3  |-  ( ph  ->  ( ( X N Y )  Fn  ( X I Y )  <-> 
( X N Y )  Fn  dom  ( X N Y ) ) )
128, 11mpbird 232 . 2  |-  ( ph  ->  ( X N Y )  Fn  ( X I Y ) )
13 df-rn 4999 . . . 4  |-  ran  ( X N Y )  =  dom  `' ( X N Y )
141, 2, 3, 4, 5invsym2 15254 . . . . . 6  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
1514dmeqd 5194 . . . . 5  |-  ( ph  ->  dom  `' ( X N Y )  =  dom  ( Y N X ) )
161, 2, 3, 5, 4, 9isoval 15256 . . . . 5  |-  ( ph  ->  ( Y I X )  =  dom  ( Y N X ) )
1715, 16eqtr4d 2498 . . . 4  |-  ( ph  ->  dom  `' ( X N Y )  =  ( Y I X ) )
1813, 17syl5eq 2507 . . 3  |-  ( ph  ->  ran  ( X N Y )  =  ( Y I X ) )
19 eqimss 3541 . . 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 5574 . 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 662 1  |-  ( ph  ->  ( X N Y ) : ( X I Y ) --> ( Y I X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823    C_ wss 3461   `'ccnv 4987   dom cdm 4988   ran crn 4989   Fun wfun 5564    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14719   Catccat 15156  Invcinv 15236    Iso ciso 15237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-cat 15160  df-cid 15161  df-sect 15238  df-inv 15239  df-iso 15240
This theorem is referenced by:  invf1o  15260  invisoinvl  15281  invcoisoid  15283  isocoinvid  15284  rcaninv  15285  ffthiso  15420  initoeu2lem1  15495
  Copyright terms: Public domain W3C validator