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Theorem invsym2 14803
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 2451 . . . . 5  |-  ( Hom  `  C )  =  ( Hom  `  C )
71, 2, 3, 4, 5, 6invss 14801 . . . 4  |-  ( ph  ->  ( Y N X )  C_  ( ( Y ( Hom  `  C
) X )  X.  ( X ( Hom  `  C ) Y ) ) )
8 relxp 5045 . . . 4  |-  Rel  (
( Y ( Hom  `  C ) X )  X.  ( X ( Hom  `  C ) Y ) )
9 relss 5025 . . . 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 18 . . 3  |-  ( ph  ->  Rel  ( Y N X ) )
11 relcnv 5304 . . 3  |-  Rel  `' ( X N Y )
1210, 11jctil 537 . 2  |-  ( ph  ->  ( Rel  `' ( X N Y )  /\  Rel  ( Y N X ) ) )
131, 2, 3, 5, 4invsym 14802 . . . 4  |-  ( ph  ->  ( f ( X N Y ) g  <-> 
g ( Y N X ) f ) )
14 vex 3071 . . . . . 6  |-  g  e. 
_V
15 vex 3071 . . . . . 6  |-  f  e. 
_V
1614, 15brcnv 5120 . . . . 5  |-  ( g `' ( X N Y ) f  <->  f ( X N Y ) g )
17 df-br 4391 . . . . 5  |-  ( g `' ( X N Y ) f  <->  <. g ,  f >.  e.  `' ( X N Y ) )
1816, 17bitr3i 251 . . . 4  |-  ( f ( X N Y ) g  <->  <. g ,  f >.  e.  `' ( X N Y ) )
19 df-br 4391 . . . 4  |-  ( g ( Y N X ) f  <->  <. g ,  f >.  e.  ( Y N X ) )
2013, 18, 193bitr3g 287 . . 3  |-  ( ph  ->  ( <. g ,  f
>.  e.  `' ( X N Y )  <->  <. g ,  f >.  e.  ( Y N X ) ) )
2120eqrelrdv2 5037 . 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 1370    e. wcel 1758    C_ wss 3426   <.cop 3981   class class class wbr 4390    X. cxp 4936   `'ccnv 4937   Rel wrel 4943   ` cfv 5516  (class class class)co 6190   Basecbs 14276   Hom chom 14351   Catccat 14704  Invcinv 14786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-sect 14788  df-inv 14789
This theorem is referenced by:  invf  14808  invf1o  14809  invinv  14810
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