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Theorem invf1o 13671
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 13670 . . 3  |-  ( ph  ->  ( X N Y ) : ( X I Y ) --> ( Y I X ) )
8 ffn 5389 . . 3  |-  ( ( X N Y ) : ( X I Y ) --> ( Y I X )  -> 
( X N Y )  Fn  ( X I Y ) )
97, 8syl 15 . 2  |-  ( ph  ->  ( X N Y )  Fn  ( X I Y ) )
101, 2, 3, 5, 4, 6invf 13670 . . . 4  |-  ( ph  ->  ( Y N X ) : ( Y I X ) --> ( X I Y ) )
11 ffn 5389 . . . 4  |-  ( ( Y N X ) : ( Y I X ) --> ( X I Y )  -> 
( Y N X )  Fn  ( Y I X ) )
1210, 11syl 15 . . 3  |-  ( ph  ->  ( Y N X )  Fn  ( Y I X ) )
131, 2, 3, 4, 5invsym2 13665 . . . 4  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
1413fneq1d 5335 . . 3  |-  ( ph  ->  ( `' ( X N Y )  Fn  ( Y I X )  <->  ( Y N X )  Fn  ( Y I X ) ) )
1512, 14mpbird 223 . 2  |-  ( ph  ->  `' ( X N Y )  Fn  ( Y I X ) )
16 dff1o4 5480 . 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 645 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 1623    e. wcel 1684   `'ccnv 4688    Fn wfn 5250   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Basecbs 13148   Catccat 13566  Invcinv 13648    Iso ciso 13649
This theorem is referenced by:  invinv  13672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-cat 13570  df-cid 13571  df-sect 13650  df-inv 13651  df-iso 13652
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