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Theorem invf 15034
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 15030 . . . 4  |-  ( ph  ->  Fun  ( X N Y ) )
7 funfn 5603 . . . 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 15031 . . . 4  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
1110fneq2d 5658 . . 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 4996 . . . 4  |-  ran  ( X N Y )  =  dom  `' ( X N Y )
141, 2, 3, 4, 5invsym2 15029 . . . . . 6  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
1514dmeqd 5191 . . . . 5  |-  ( ph  ->  dom  `' ( X N Y )  =  dom  ( Y N X ) )
161, 2, 3, 5, 4, 9isoval 15031 . . . . 5  |-  ( ph  ->  ( Y I X )  =  dom  ( Y N X ) )
1715, 16eqtr4d 2485 . . . 4  |-  ( ph  ->  dom  `' ( X N Y )  =  ( Y I X ) )
1813, 17syl5eq 2494 . . 3  |-  ( ph  ->  ran  ( X N Y )  =  ( Y I X ) )
19 eqimss 3538 . . 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 5578 . 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 1381    e. wcel 1802    C_ wss 3458   `'ccnv 4984   dom cdm 4985   ran crn 4986   Fun wfun 5568    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277   Basecbs 14504   Catccat 14933  Invcinv 15012    Iso ciso 15013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-cat 14937  df-cid 14938  df-sect 15014  df-inv 15015  df-iso 15016
This theorem is referenced by:  invf1o  15035  ffthiso  15167
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