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Theorem brcic 16458
Description: The relation "is isomorphic to" for categories. (Contributed by AV, 5-Apr-2020.)
Hypotheses
Ref Expression
cic.i |- I = ( Iso ` C )
cic.b |- B = ( Base ` C )
cic.c |- ( ph -> C e. Cat )
cic.x |- ( ph -> X e. B )
cic.y |- ( ph -> Y e. B )
Assertion
Ref Expression
brcic |- ( ph -> ( X ( ~=c𝑐 ` C ) Y <-> ( X I Y ) =/= (/) ) )
Proof of Theorem brcic
StepHypRef Expression
1 cic.c . . . 4 |- ( ph -> C e. Cat )
2 cicfval 16457 . . . 4 |- ( C e. Cat -> ( ~=c𝑐 ` C ) = ( ( Iso ` C ) supp (/) ) )
31, 2syl 17 . . 3 |- ( ph -> ( ~=c𝑐 ` C ) = ( ( Iso ` C ) supp (/) ) )
43breqd 4664 . 2 |- ( ph -> ( X ( ~=c𝑐 ` C ) Y <-> X ( ( Iso ` C ) supp (/) ) Y ) )
5 df-br 4654 . . 3 |- ( X ( ( Iso ` C ) supp (/) ) Y <-> <. X , Y >. e. ( ( Iso ` C ) supp (/) ) )
65a1i 11 . 2 |- ( ph -> ( X ( ( Iso ` C ) supp (/) ) Y <-> <. X , Y >. e. ( ( Iso ` C ) supp (/) ) ) )
7 cic.i . . . . . 6 |- I = ( Iso ` C )
87a1i 11 . . . . 5 |- ( ph -> I = ( Iso ` C ) )
98fveq1d 6193 . . . 4 |- ( ph -> ( I ` <. X , Y >. ) = ( ( Iso ` C ) ` <. X , Y >. ) )
109neeq1d 2853 . . 3 |- ( ph -> ( ( I ` <. X , Y >. ) =/= (/) <-> ( ( Iso ` C ) ` <. X , Y >. ) =/= (/) ) )
11 df-ov 6653 . . . . . 6 |- ( X I Y ) = ( I ` <. X , Y >. )
1211eqcomi 2631 . . . . 5 |- ( I ` <. X , Y >. ) = ( X I Y )
1312a1i 11 . . . 4 |- ( ph -> ( I ` <. X , Y >. ) = ( X I Y ) )
1413neeq1d 2853 . . 3 |- ( ph -> ( ( I ` <. X , Y >. ) =/= (/) <-> ( X I Y ) =/= (/) ) )
15 fvexd 6203 . . . . 5 |- ( ph -> ( Base ` C ) e. _V )
16 sqxpexg 6963 . . . . 5 |- ( ( Base ` C ) e. _V -> ( ( Base ` C ) X. ( Base ` C ) ) e. _V )
1715, 16syl 17 . . . 4 |- ( ph -> ( ( Base ` C ) X. ( Base ` C ) ) e. _V )
18 cic.x . . . . . 6 |- ( ph -> X e. B )
19 cic.b . . . . . 6 |- B = ( Base ` C )
2018, 19syl6eleq 2711 . . . . 5 |- ( ph -> X e. ( Base ` C ) )
21 cic.y . . . . . 6 |- ( ph -> Y e. B )
2221, 19syl6eleq 2711 . . . . 5 |- ( ph -> Y e. ( Base ` C ) )
23 opelxp 5146 . . . . 5 |- ( <. X , Y >. e. ( ( Base ` C ) X. ( Base ` C ) ) <-> ( X e. ( Base ` C ) /\ Y e. ( Base ` C ) ) )
2420, 22, 23sylanbrc 698 . . . 4 |- ( ph -> <. X , Y >. e. ( ( Base ` C ) X. ( Base ` C ) ) )
25 isofn 16435 . . . . 5 |- ( C e. Cat -> ( Iso ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
261, 25syl 17 . . . 4 |- ( ph -> ( Iso ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
27 fvn0elsuppb 7312 . . . 4 |- ( ( ( ( Base ` C ) X. ( Base ` C ) ) e. _V /\ <. X , Y >. e. ( ( Base ` C ) X. ( Base ` C ) ) /\ ( Iso ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) ) -> ( ( ( Iso ` C ) ` <. X , Y >. ) =/= (/) <-> <. X , Y >. e. ( ( Iso ` C ) supp (/) ) ) )
2817, 24, 26, 27syl3anc 1326 . . 3 |- ( ph -> ( ( ( Iso ` C ) ` <. X , Y >. ) =/= (/) <-> <. X , Y >. e. ( ( Iso ` C ) supp (/) ) ) )
2910, 14, 283bitr3rd 299 . 2 |- ( ph -> ( <. X , Y >. e. ( ( Iso ` C ) supp (/) ) <-> ( X I Y ) =/= (/) ) )
304, 6, 293bitrd 294 1 |- ( ph -> ( X ( ~=c𝑐 ` C ) Y <-> ( X I Y ) =/= (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   <.cop 4183   class class class wbr 4653   X. cxp 5112   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Basecbs 15857   Catccat 16325   Iso ciso 16406   ~=c𝑐 ccic 16455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-supp 7296  df-inv 16408  df-iso 16409  df-cic 16456
This theorem is referenced by:  cic  16459
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