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Theorem cicer 15789
Description: Isomorphism is an equivalence relation on objects of a category. Remark 3.16 in [Adamek] p. 29. (Contributed by AV, 5-Apr-2020.)
Assertion
Ref Expression
cicer  |-  ( C  e.  Cat  ->  (  ~=c𝑐  `  C )  Er  ( Base `  C ) )

Proof of Theorem cicer
Dummy variables  f  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 4965 . . . . . 6  |-  Rel  { <. x ,  y >.  |  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  ( (  Iso  `  C
) `  <. x ,  y >. )  =/=  (/) ) }
21a1i 11 . . . . 5  |-  ( C  e.  Cat  ->  Rel  {
<. x ,  y >.  |  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  ( (  Iso  `  C
) `  <. x ,  y >. )  =/=  (/) ) } )
3 fveq2 5879 . . . . . . . . 9  |-  ( f  =  <. x ,  y
>.  ->  ( (  Iso  `  C ) `  f
)  =  ( (  Iso  `  C ) `  <. x ,  y
>. ) )
43neeq1d 2702 . . . . . . . 8  |-  ( f  =  <. x ,  y
>.  ->  ( ( (  Iso  `  C ) `  f )  =/=  (/)  <->  ( (  Iso  `  C ) `  <. x ,  y >.
)  =/=  (/) ) )
54rabxp 4876 . . . . . . 7  |-  { f  e.  ( ( Base `  C )  X.  ( Base `  C ) )  |  ( (  Iso  `  C ) `  f
)  =/=  (/) }  =  { <. x ,  y
>.  |  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  ( (  Iso  `  C
) `  <. x ,  y >. )  =/=  (/) ) }
65a1i 11 . . . . . 6  |-  ( C  e.  Cat  ->  { f  e.  ( ( Base `  C )  X.  ( Base `  C ) )  |  ( (  Iso  `  C ) `  f
)  =/=  (/) }  =  { <. x ,  y
>.  |  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  ( (  Iso  `  C
) `  <. x ,  y >. )  =/=  (/) ) } )
76releqd 4924 . . . . 5  |-  ( C  e.  Cat  ->  ( Rel  { f  e.  ( ( Base `  C
)  X.  ( Base `  C ) )  |  ( (  Iso  `  C
) `  f )  =/=  (/) }  <->  Rel  { <. x ,  y >.  |  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C
)  /\  ( (  Iso  `  C ) `  <. x ,  y >.
)  =/=  (/) ) } ) )
82, 7mpbird 240 . . . 4  |-  ( C  e.  Cat  ->  Rel  { f  e.  ( (
Base `  C )  X.  ( Base `  C
) )  |  ( (  Iso  `  C
) `  f )  =/=  (/) } )
9 isofn 15758 . . . . . 6  |-  ( C  e.  Cat  ->  (  Iso  `  C )  Fn  ( ( Base `  C
)  X.  ( Base `  C ) ) )
10 fvex 5889 . . . . . . 7  |-  ( Base `  C )  e.  _V
11 sqxpexg 6615 . . . . . . 7  |-  ( (
Base `  C )  e.  _V  ->  ( ( Base `  C )  X.  ( Base `  C
) )  e.  _V )
1210, 11mp1i 13 . . . . . 6  |-  ( C  e.  Cat  ->  (
( Base `  C )  X.  ( Base `  C
) )  e.  _V )
13 0ex 4528 . . . . . . 7  |-  (/)  e.  _V
1413a1i 11 . . . . . 6  |-  ( C  e.  Cat  ->  (/)  e.  _V )
15 suppvalfn 6940 . . . . . 6  |-  ( ( (  Iso  `  C
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) )  /\  (
( Base `  C )  X.  ( Base `  C
) )  e.  _V  /\  (/)  e.  _V )  -> 
( (  Iso  `  C
) supp  (/) )  =  {
f  e.  ( (
Base `  C )  X.  ( Base `  C
) )  |  ( (  Iso  `  C
) `  f )  =/=  (/) } )
169, 12, 14, 15syl3anc 1292 . . . . 5  |-  ( C  e.  Cat  ->  (
(  Iso  `  C ) supp  (/) )  =  {
f  e.  ( (
Base `  C )  X.  ( Base `  C
) )  |  ( (  Iso  `  C
) `  f )  =/=  (/) } )
1716releqd 4924 . . . 4  |-  ( C  e.  Cat  ->  ( Rel  ( (  Iso  `  C
) supp  (/) )  <->  Rel  { f  e.  ( ( Base `  C )  X.  ( Base `  C ) )  |  ( (  Iso  `  C ) `  f
)  =/=  (/) } ) )
188, 17mpbird 240 . . 3  |-  ( C  e.  Cat  ->  Rel  ( (  Iso  `  C
) supp  (/) ) )
19 cicfval 15780 . . . 4  |-  ( C  e.  Cat  ->  (  ~=c𝑐  `  C )  =  ( (  Iso  `  C
) supp  (/) ) )
2019releqd 4924 . . 3  |-  ( C  e.  Cat  ->  ( Rel  (  ~=c𝑐  `  C )  <->  Rel  ( (  Iso  `  C ) supp  (/) ) ) )
2118, 20mpbird 240 . 2  |-  ( C  e.  Cat  ->  Rel  (  ~=c𝑐  `  C ) )
22 cicsym 15787 . 2  |-  ( ( C  e.  Cat  /\  x (  ~=c𝑐  `  C ) y )  ->  y
(  ~=c𝑐  `  C ) x )
23 cictr 15788 . . 3  |-  ( ( C  e.  Cat  /\  x (  ~=c𝑐  `  C ) y  /\  y ( 
~=c𝑐  `  C ) z )  ->  x (  ~=c𝑐  `  C
) z )
24233expb 1232 . 2  |-  ( ( C  e.  Cat  /\  ( x (  ~=c𝑐  `  C
) y  /\  y
(  ~=c𝑐  `  C ) z ) )  ->  x (  ~=c𝑐  `  C ) z )
25 cicref 15784 . . 3  |-  ( ( C  e.  Cat  /\  x  e.  ( Base `  C ) )  ->  x (  ~=c𝑐  `  C ) x )
26 ciclcl 15785 . . 3  |-  ( ( C  e.  Cat  /\  x (  ~=c𝑐  `  C ) x )  ->  x  e.  ( Base `  C
) )
2725, 26impbida 850 . 2  |-  ( C  e.  Cat  ->  (
x  e.  ( Base `  C )  <->  x (  ~=c𝑐  `  C ) x ) )
2821, 22, 24, 27iserd 7407 1  |-  ( C  e.  Cat  ->  (  ~=c𝑐  `  C )  Er  ( Base `  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   {crab 2760   _Vcvv 3031   (/)c0 3722   <.cop 3965   class class class wbr 4395   {copab 4453    X. cxp 4837   Rel wrel 4844    Fn wfn 5584   ` cfv 5589  (class class class)co 6308   supp csupp 6933    Er wer 7378   Basecbs 15199   Catccat 15648    Iso ciso 15729    ~=c𝑐 ccic 15778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-supp 6934  df-er 7381  df-cat 15652  df-cid 15653  df-sect 15730  df-inv 15731  df-iso 15732  df-cic 15779
This theorem is referenced by: (None)
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