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Theorem inviso2 15013
Description: If  G is an inverse to  F, then  G is an isomorphism. (Contributed by Mario Carneiro, 3-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
)
inviso1.1  |-  ( ph  ->  F ( X N Y ) G )
Assertion
Ref Expression
inviso2  |-  ( ph  ->  G  e.  ( Y I X ) )

Proof of Theorem inviso2
StepHypRef Expression
1 invfval.b . 2  |-  B  =  ( Base `  C
)
2 invfval.n . 2  |-  N  =  (Inv `  C )
3 invfval.c . 2  |-  ( ph  ->  C  e.  Cat )
4 invfval.y . 2  |-  ( ph  ->  Y  e.  B )
5 invfval.x . 2  |-  ( ph  ->  X  e.  B )
6 isoval.n . 2  |-  I  =  (  Iso  `  C
)
7 inviso1.1 . . 3  |-  ( ph  ->  F ( X N Y ) G )
81, 2, 3, 5, 4invsym 15008 . . 3  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
G ( Y N X ) F ) )
97, 8mpbid 210 . 2  |-  ( ph  ->  G ( Y N X ) F )
101, 2, 3, 4, 5, 6, 9inviso1 15012 1  |-  ( ph  ->  G  e.  ( Y I X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762   class class class wbr 4442   ` 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:  yonffthlem  15400
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