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Theorem invf 14810
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 14806 . . . 4  |-  ( ph  ->  Fun  ( X N Y ) )
7 funfn 5547 . . . 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 14807 . . . 4  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
1110fneq2d 5602 . . 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 4951 . . . 4  |-  ran  ( X N Y )  =  dom  `' ( X N Y )
141, 2, 3, 4, 5invsym2 14805 . . . . . 6  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
1514dmeqd 5142 . . . . 5  |-  ( ph  ->  dom  `' ( X N Y )  =  dom  ( Y N X ) )
161, 2, 3, 5, 4, 9isoval 14807 . . . . 5  |-  ( ph  ->  ( Y I X )  =  dom  ( Y N X ) )
1715, 16eqtr4d 2495 . . . 4  |-  ( ph  ->  dom  `' ( X N Y )  =  ( Y I X ) )
1813, 17syl5eq 2504 . . 3  |-  ( ph  ->  ran  ( X N Y )  =  ( Y I X ) )
19 eqimss 3508 . . 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 5522 . 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 664 1  |-  ( ph  ->  ( X N Y ) : ( X I Y ) --> ( Y I X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    C_ wss 3428   `'ccnv 4939   dom cdm 4940   ran crn 4941   Fun wfun 5512    Fn wfn 5513   -->wf 5514   ` cfv 5518  (class class class)co 6192   Basecbs 14278   Catccat 14706  Invcinv 14788    Iso ciso 14789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-cat 14710  df-cid 14711  df-sect 14790  df-inv 14791  df-iso 14792
This theorem is referenced by:  invf1o  14811  ffthiso  14943
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