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Theorem invsym2 13981
Description: The inverse relation is symmetric. (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 )
Assertion
Ref Expression
invsym2  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )

Proof of Theorem invsym2
Dummy variables  f 
g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfval.b . . . . 5  |-  B  =  ( Base `  C
)
2 invfval.n . . . . 5  |-  N  =  (Inv `  C )
3 invfval.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
4 invfval.y . . . . 5  |-  ( ph  ->  Y  e.  B )
5 invfval.x . . . . 5  |-  ( ph  ->  X  e.  B )
6 eqid 2436 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
71, 2, 3, 4, 5, 6invss 13979 . . . 4  |-  ( ph  ->  ( Y N X )  C_  ( ( Y (  Hom  `  C
) X )  X.  ( X (  Hom  `  C ) Y ) ) )
8 relxp 4976 . . . 4  |-  Rel  (
( Y (  Hom  `  C ) X )  X.  ( X (  Hom  `  C ) Y ) )
9 relss 4956 . . . 4  |-  ( ( Y N X ) 
C_  ( ( Y (  Hom  `  C
) X )  X.  ( X (  Hom  `  C ) Y ) )  ->  ( Rel  ( ( Y (  Hom  `  C ) X )  X.  ( X (  Hom  `  C
) Y ) )  ->  Rel  ( Y N X ) ) )
107, 8, 9ee10 1385 . . 3  |-  ( ph  ->  Rel  ( Y N X ) )
11 relcnv 5235 . . 3  |-  Rel  `' ( X N Y )
1210, 11jctil 524 . 2  |-  ( ph  ->  ( Rel  `' ( X N Y )  /\  Rel  ( Y N X ) ) )
131, 2, 3, 5, 4invsym 13980 . . . 4  |-  ( ph  ->  ( f ( X N Y ) g  <-> 
g ( Y N X ) f ) )
14 vex 2952 . . . . . 6  |-  g  e. 
_V
15 vex 2952 . . . . . 6  |-  f  e. 
_V
1614, 15brcnv 5048 . . . . 5  |-  ( g `' ( X N Y ) f  <->  f ( X N Y ) g )
17 df-br 4206 . . . . 5  |-  ( g `' ( X N Y ) f  <->  <. g ,  f >.  e.  `' ( X N Y ) )
1816, 17bitr3i 243 . . . 4  |-  ( f ( X N Y ) g  <->  <. g ,  f >.  e.  `' ( X N Y ) )
19 df-br 4206 . . . 4  |-  ( g ( Y N X ) f  <->  <. g ,  f >.  e.  ( Y N X ) )
2013, 18, 193bitr3g 279 . . 3  |-  ( ph  ->  ( <. g ,  f
>.  e.  `' ( X N Y )  <->  <. g ,  f >.  e.  ( Y N X ) ) )
2120eqrelrdv2 4968 . 2  |-  ( ( ( Rel  `' ( X N Y )  /\  Rel  ( Y N X ) )  /\  ph )  ->  `' ( X N Y )  =  ( Y N X ) )
2212, 21mpancom 651 1  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3313   <.cop 3810   class class class wbr 4205    X. cxp 4869   `'ccnv 4870   Rel wrel 4876   ` cfv 5447  (class class class)co 6074   Basecbs 13462    Hom chom 13533   Catccat 13882  Invcinv 13964
This theorem is referenced by:  invf  13986  invf1o  13987  invinv  13988
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-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-sect 13966  df-inv 13967
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