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Theorem invf1o 13986
Description: The inverse relation is a bijection 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
invf1o  |-  ( ph  ->  ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) )

Proof of Theorem invf1o
StepHypRef Expression
1 invfval.b . . . 4  |-  B  =  ( Base `  C
)
2 invfval.n . . . 4  |-  N  =  (Inv `  C )
3 invfval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
4 invfval.x . . . 4  |-  ( ph  ->  X  e.  B )
5 invfval.y . . . 4  |-  ( ph  ->  Y  e.  B )
6 isoval.n . . . 4  |-  I  =  (  Iso  `  C
)
71, 2, 3, 4, 5, 6invf 13985 . . 3  |-  ( ph  ->  ( X N Y ) : ( X I Y ) --> ( Y I X ) )
8 ffn 5583 . . 3  |-  ( ( X N Y ) : ( X I Y ) --> ( Y I X )  -> 
( X N Y )  Fn  ( X I Y ) )
97, 8syl 16 . 2  |-  ( ph  ->  ( X N Y )  Fn  ( X I Y ) )
101, 2, 3, 5, 4, 6invf 13985 . . . 4  |-  ( ph  ->  ( Y N X ) : ( Y I X ) --> ( X I Y ) )
11 ffn 5583 . . . 4  |-  ( ( Y N X ) : ( Y I X ) --> ( X I Y )  -> 
( Y N X )  Fn  ( Y I X ) )
1210, 11syl 16 . . 3  |-  ( ph  ->  ( Y N X )  Fn  ( Y I X ) )
131, 2, 3, 4, 5invsym2 13980 . . . 4  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
1413fneq1d 5528 . . 3  |-  ( ph  ->  ( `' ( X N Y )  Fn  ( Y I X )  <->  ( Y N X )  Fn  ( Y I X ) ) )
1512, 14mpbird 224 . 2  |-  ( ph  ->  `' ( X N Y )  Fn  ( Y I X ) )
16 dff1o4 5674 . 2  |-  ( ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X )  <->  ( ( X N Y )  Fn  ( X I Y )  /\  `' ( X N Y )  Fn  ( Y I X ) ) )
179, 15, 16sylanbrc 646 1  |-  ( ph  ->  ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   `'ccnv 4869    Fn wfn 5441   -->wf 5442   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   Basecbs 13461   Catccat 13881  Invcinv 13963    Iso ciso 13964
This theorem is referenced by:  invinv  13987
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-cat 13885  df-cid 13886  df-sect 13965  df-inv 13966  df-iso 13967
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