Theorem ciclcl 16462

Description: Isomorphism implies the left side is an object. (Contributed by AV, 5-Apr-2020.)

Assertion

Ref	Expression
ciclcl	|- ( ( C e. Cat /\ R ( ~=c𝑐 ` C ) S ) -> R e. ( Base ` C ) )

Proof of Theorem ciclcl

Step	Hyp	Ref	Expression
1		cicfval 16457	. . . 4 |- ( C e. Cat -> ( ~=c𝑐 ` C ) = ( ( Iso ` C ) supp (/) ) )
2	1	breqd 4664	. . 3 |- ( C e. Cat -> ( R ( ~=c𝑐 ` C ) S <-> R ( ( Iso ` C ) supp (/) ) S ) )
3		isofn 16435	. . . . 5 |- ( C e. Cat -> ( Iso ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
4		fvex 6201	. . . . . 6 |- ( Base ` C ) e. _V
5		sqxpexg 6963	. . . . . 6 |- ( ( Base ` C ) e. _V -> ( ( Base ` C ) X. ( Base ` C ) ) e. _V )
6	4, 5	mp1i 13	. . . . 5 |- ( C e. Cat -> ( ( Base ` C ) X. ( Base ` C ) ) e. _V )
7		0ex 4790	. . . . . 6 |- (/) e. _V
8	7	a1i 11	. . . . 5 |- ( C e. Cat -> (/) e. _V )
9		df-br 4654	. . . . . 6 |- ( R ( ( Iso ` C ) supp (/) ) S <-> <. R , S >. e. ( ( Iso ` C ) supp (/) ) )
10		elsuppfn 7303	. . . . . 6 |- ( ( ( Iso ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) /\ ( ( Base ` C ) X. ( Base ` C ) ) e. _V /\ (/) e. _V ) -> ( <. R , S >. e. ( ( Iso ` C ) supp (/) ) <-> ( <. R , S >. e. ( ( Base ` C ) X. ( Base ` C ) ) /\ ( ( Iso ` C ) ` <. R , S >. ) =/= (/) ) ) )
11	9, 10	syl5bb 272	. . . . 5 |- ( ( ( Iso ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) /\ ( ( Base ` C ) X. ( Base ` C ) ) e. _V /\ (/) e. _V ) -> ( R ( ( Iso ` C ) supp (/) ) S <-> ( <. R , S >. e. ( ( Base ` C ) X. ( Base ` C ) ) /\ ( ( Iso ` C ) ` <. R , S >. ) =/= (/) ) ) )
12	3, 6, 8, 11	syl3anc 1326	. . . 4 |- ( C e. Cat -> ( R ( ( Iso ` C ) supp (/) ) S <-> ( <. R , S >. e. ( ( Base ` C ) X. ( Base ` C ) ) /\ ( ( Iso ` C ) ` <. R , S >. ) =/= (/) ) ) )
13		opelxp1 5150	. . . . 5 |- ( <. R , S >. e. ( ( Base ` C ) X. ( Base ` C ) ) -> R e. ( Base ` C ) )
14	13	adantr 481	. . . 4 |- ( ( <. R , S >. e. ( ( Base ` C ) X. ( Base ` C ) ) /\ ( ( Iso ` C ) ` <. R , S >. ) =/= (/) ) -> R e. ( Base ` C ) )
15	12, 14	syl6bi 243	. . 3 |- ( C e. Cat -> ( R ( ( Iso ` C ) supp (/) ) S -> R e. ( Base ` C ) ) )
16	2, 15	sylbid 230	. 2 |- ( C e. Cat -> ( R ( ~=c𝑐 ` C ) S -> R e. ( Base ` C ) ) )
17	16	imp 445	1 |- ( ( C e. Cat /\ R ( ~=c𝑐 ` C ) S ) -> R e. ( Base ` C ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   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   ~=c_𝑐 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:  cicsym  16464  cictr  16465  cicer  16466  initoeu2  16666