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Theorem invf 13719
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 13715 . . . 4  |-  ( ph  ->  Fun  ( X N Y ) )
7 funfn 5320 . . . 4  |-  ( Fun  ( X N Y )  <->  ( X N Y )  Fn  dom  ( X N Y ) )
86, 7sylib 188 . . 3  |-  ( ph  ->  ( X N Y )  Fn  dom  ( X N Y ) )
9 isoval.n . . . . 5  |-  I  =  (  Iso  `  C
)
101, 2, 3, 4, 5, 9isoval 13716 . . . 4  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
1110fneq2d 5373 . . 3  |-  ( ph  ->  ( ( X N Y )  Fn  ( X I Y )  <-> 
( X N Y )  Fn  dom  ( X N Y ) ) )
128, 11mpbird 223 . 2  |-  ( ph  ->  ( X N Y )  Fn  ( X I Y ) )
13 df-rn 4737 . . . 4  |-  ran  ( X N Y )  =  dom  `' ( X N Y )
141, 2, 3, 4, 5invsym2 13714 . . . . . 6  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
1514dmeqd 4918 . . . . 5  |-  ( ph  ->  dom  `' ( X N Y )  =  dom  ( Y N X ) )
161, 2, 3, 5, 4, 9isoval 13716 . . . . 5  |-  ( ph  ->  ( Y I X )  =  dom  ( Y N X ) )
1715, 16eqtr4d 2351 . . . 4  |-  ( ph  ->  dom  `' ( X N Y )  =  ( Y I X ) )
1813, 17syl5eq 2360 . . 3  |-  ( ph  ->  ran  ( X N Y )  =  ( Y I X ) )
19 eqimss 3264 . . 3  |-  ( ran  ( X N Y )  =  ( Y I X )  ->  ran  ( X N Y )  C_  ( Y I X ) )
2018, 19syl 15 . 2  |-  ( ph  ->  ran  ( X N Y )  C_  ( Y I X ) )
21 df-f 5296 . 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 645 1  |-  ( ph  ->  ( X N Y ) : ( X I Y ) --> ( Y I X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701    C_ wss 3186   `'ccnv 4725   dom cdm 4726   ran crn 4727   Fun wfun 5286    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900   Basecbs 13195   Catccat 13615  Invcinv 13697    Iso ciso 13698
This theorem is referenced by:  invf1o  13720  ffthiso  13852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-cat 13619  df-cid 13620  df-sect 13699  df-inv 13700  df-iso 13701
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