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Theorem invf1o 15015
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 15014 . . 3  |-  ( ph  ->  ( X N Y ) : ( X I Y ) --> ( Y I X ) )
8 ffn 5724 . . 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 15014 . . . 4  |-  ( ph  ->  ( Y N X ) : ( Y I X ) --> ( X I Y ) )
11 ffn 5724 . . . 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 15009 . . . 4  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
1413fneq1d 5664 . . 3  |-  ( ph  ->  ( `' ( X N Y )  Fn  ( Y I X )  <->  ( Y N X )  Fn  ( Y I X ) ) )
1512, 14mpbird 232 . 2  |-  ( ph  ->  `' ( X N Y )  Fn  ( Y I X ) )
16 dff1o4 5817 . 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 664 1  |-  ( ph  ->  ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   `'ccnv 4993    Fn wfn 5576   -->wf 5577   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6277   Basecbs 14481   Catccat 14910  Invcinv 14992    Iso ciso 14993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-cat 14914  df-cid 14915  df-sect 14994  df-inv 14995  df-iso 14996
This theorem is referenced by:  invinv  15016
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