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Theorem invsym2 15378
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 2404 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
71, 2, 3, 4, 5, 6invss 15376 . . . 4  |-  ( ph  ->  ( Y N X )  C_  ( ( Y ( Hom  `  C
) X )  X.  ( X ( Hom  `  C ) Y ) ) )
8 relxp 4933 . . . 4  |-  Rel  (
( Y ( Hom  `  C ) X )  X.  ( X ( Hom  `  C ) Y ) )
9 relss 4913 . . . 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 5197 . . 3  |-  Rel  `' ( X N Y )
1210, 11jctil 537 . 2  |-  ( ph  ->  ( Rel  `' ( X N Y )  /\  Rel  ( Y N X ) ) )
131, 2, 3, 5, 4invsym 15377 . . . 4  |-  ( ph  ->  ( f ( X N Y ) g  <-> 
g ( Y N X ) f ) )
14 vex 3064 . . . . . 6  |-  g  e. 
_V
15 vex 3064 . . . . . 6  |-  f  e. 
_V
1614, 15brcnv 5008 . . . . 5  |-  ( g `' ( X N Y ) f  <->  f ( X N Y ) g )
17 df-br 4398 . . . . 5  |-  ( g `' ( X N Y ) f  <->  <. g ,  f >.  e.  `' ( X N Y ) )
1816, 17bitr3i 253 . . . 4  |-  ( f ( X N Y ) g  <->  <. g ,  f >.  e.  `' ( X N Y ) )
19 df-br 4398 . . . 4  |-  ( g ( Y N X ) f  <->  <. g ,  f >.  e.  ( Y N X ) )
2013, 18, 193bitr3g 289 . . 3  |-  ( ph  ->  ( <. g ,  f
>.  e.  `' ( X N Y )  <->  <. g ,  f >.  e.  ( Y N X ) ) )
2120eqrelrdv2 4925 . 2  |-  ( ( ( Rel  `' ( X N Y )  /\  Rel  ( Y N X ) )  /\  ph )  ->  `' ( X N Y )  =  ( Y N X ) )
2212, 21mpancom 669 1  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    e. wcel 1844    C_ wss 3416   <.cop 3980   class class class wbr 4397    X. cxp 4823   `'ccnv 4824   Rel wrel 4830   ` cfv 5571  (class class class)co 6280   Basecbs 14843   Hom chom 14922   Catccat 15280  Invcinv 15360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-sect 15362  df-inv 15363
This theorem is referenced by:  invf  15383  invf1o  15384  invinv  15385  cicsym  15419
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