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Theorem inviso2 13913
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 13908 . . 3  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
G ( Y N X ) F ) )
97, 8mpbid 202 . 2  |-  ( ph  ->  G ( Y N X ) F )
101, 2, 3, 4, 5, 6, 9inviso1 13912 1  |-  ( ph  ->  G  e.  ( Y I X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   class class class wbr 4147   ` cfv 5388  (class class class)co 6014   Basecbs 13390   Catccat 13810  Invcinv 13892    Iso ciso 13893
This theorem is referenced by:  yonffthlem  14300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-reu 2650  df-rmo 2651  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-id 4433  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-ov 6017  df-oprab 6018  df-mpt2 6019  df-1st 6282  df-2nd 6283  df-riota 6479  df-cat 13814  df-cid 13815  df-sect 13894  df-inv 13895  df-iso 13896
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