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Theorem invsym2 15661
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 2423 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
71, 2, 3, 4, 5, 6invss 15659 . . . 4  |-  ( ph  ->  ( Y N X )  C_  ( ( Y ( Hom  `  C
) X )  X.  ( X ( Hom  `  C ) Y ) ) )
8 relxp 4959 . . . 4  |-  Rel  (
( Y ( Hom  `  C ) X )  X.  ( X ( Hom  `  C ) Y ) )
9 relss 4939 . . . 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, 9mpisyl 22 . . 3  |-  ( ph  ->  Rel  ( Y N X ) )
11 relcnv 5224 . . 3  |-  Rel  `' ( X N Y )
1210, 11jctil 540 . 2  |-  ( ph  ->  ( Rel  `' ( X N Y )  /\  Rel  ( Y N X ) ) )
131, 2, 3, 5, 4invsym 15660 . . . 4  |-  ( ph  ->  ( f ( X N Y ) g  <-> 
g ( Y N X ) f ) )
14 vex 3085 . . . . . 6  |-  g  e. 
_V
15 vex 3085 . . . . . 6  |-  f  e. 
_V
1614, 15brcnv 5034 . . . . 5  |-  ( g `' ( X N Y ) f  <->  f ( X N Y ) g )
17 df-br 4422 . . . . 5  |-  ( g `' ( X N Y ) f  <->  <. g ,  f >.  e.  `' ( X N Y ) )
1816, 17bitr3i 255 . . . 4  |-  ( f ( X N Y ) g  <->  <. g ,  f >.  e.  `' ( X N Y ) )
19 df-br 4422 . . . 4  |-  ( g ( Y N X ) f  <->  <. g ,  f >.  e.  ( Y N X ) )
2013, 18, 193bitr3g 291 . . 3  |-  ( ph  ->  ( <. g ,  f
>.  e.  `' ( X N Y )  <->  <. g ,  f >.  e.  ( Y N X ) ) )
2120eqrelrdv2 4951 . 2  |-  ( ( ( Rel  `' ( X N Y )  /\  Rel  ( Y N X ) )  /\  ph )  ->  `' ( X N Y )  =  ( Y N X ) )
2212, 21mpancom 674 1  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1438    e. wcel 1869    C_ wss 3437   <.cop 4003   class class class wbr 4421    X. cxp 4849   `'ccnv 4850   Rel wrel 4856   ` cfv 5599  (class class class)co 6303   Basecbs 15114   Hom chom 15194   Catccat 15563  Invcinv 15643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-sect 15645  df-inv 15646
This theorem is referenced by:  invf  15666  invf1o  15667  invinv  15668  cicsym  15702
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