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Theorem invsym2 13908
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 2380 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
71, 2, 3, 4, 5, 6invss 13906 . . . 4  |-  ( ph  ->  ( Y N X )  C_  ( ( Y (  Hom  `  C
) X )  X.  ( X (  Hom  `  C ) Y ) ) )
8 relxp 4916 . . . 4  |-  Rel  (
( Y (  Hom  `  C ) X )  X.  ( X (  Hom  `  C ) Y ) )
9 relss 4896 . . . 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 1382 . . 3  |-  ( ph  ->  Rel  ( Y N X ) )
11 relcnv 5175 . . 3  |-  Rel  `' ( X N Y )
1210, 11jctil 524 . 2  |-  ( ph  ->  ( Rel  `' ( X N Y )  /\  Rel  ( Y N X ) ) )
131, 2, 3, 5, 4invsym 13907 . . . 4  |-  ( ph  ->  ( f ( X N Y ) g  <-> 
g ( Y N X ) f ) )
14 vex 2895 . . . . . 6  |-  g  e. 
_V
15 vex 2895 . . . . . 6  |-  f  e. 
_V
1614, 15brcnv 4988 . . . . 5  |-  ( g `' ( X N Y ) f  <->  f ( X N Y ) g )
17 df-br 4147 . . . . 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 4147 . . . 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 4908 . 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 1649    e. wcel 1717    C_ wss 3256   <.cop 3753   class class class wbr 4146    X. cxp 4809   `'ccnv 4810   Rel wrel 4816   ` cfv 5387  (class class class)co 6013   Basecbs 13389    Hom chom 13460   Catccat 13809  Invcinv 13891
This theorem is referenced by:  invf  13913  invf1o  13914  invinv  13915
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-sect 13893  df-inv 13894
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