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Theorem inviso2 13985
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 13980 . . 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 13984 1  |-  ( ph  ->  G  e.  ( Y I X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   class class class wbr 4205   ` cfv 5447  (class class class)co 6074   Basecbs 13462   Catccat 13882  Invcinv 13964    Iso ciso 13965
This theorem is referenced by:  yonffthlem  14372
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 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-cat 13886  df-cid 13887  df-sect 13966  df-inv 13967  df-iso 13968
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