Theorem cicref 16461

Description: Isomorphism is reflexive. (Contributed by AV, 5-Apr-2020.)

Assertion

Ref	Expression
cicref	|- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> O ( ~=c𝑐 ` C ) O )

Proof of Theorem cicref

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- ( Iso ` C ) = ( Iso ` C )
2		eqid 2622	. 2 |- ( Base ` C ) = ( Base ` C )
3		simpl 473	. 2 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> C e. Cat )
4		simpr 477	. 2 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> O e. ( Base ` C ) )
5		eqid 2622	. . 3 |- (Inv ` C ) = (Inv ` C )
6		eqid 2622	. . . . . 6 |- ( Hom ` C ) = ( Hom ` C )
7		eqid 2622	. . . . . 6 |- ( Id ` C ) = ( Id ` C )
8		eqid 2622	. . . . . 6 |- (comp ` C ) = (comp ` C )
9	2, 6, 7, 3, 4	catidcl 16343	. . . . . 6 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> ( ( Id ` C ) ` O ) e. ( O ( Hom ` C ) O ) )
10	2, 6, 7, 3, 4, 8, 4, 9	catrid 16345	. . . . 5 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> ( ( ( Id ` C ) ` O ) ( <. O , O >. (comp ` C ) O ) ( ( Id ` C ) ` O ) ) = ( ( Id ` C ) ` O ) )
11		eqid 2622	. . . . . . 7 |- (Sect ` C ) = (Sect ` C )
12	2, 6, 8, 7, 11, 3, 4, 4, 9, 9	issect2 16414	. . . . . 6 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> ( ( ( Id ` C ) ` O ) ( O (Sect ` C ) O ) ( ( Id ` C ) ` O ) <-> ( ( ( Id ` C ) ` O ) ( <. O , O >. (comp ` C ) O ) ( ( Id ` C ) ` O ) ) = ( ( Id ` C ) ` O ) ) )
13	12, 12	anbi12d 747	. . . . 5 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> ( ( ( ( Id ` C ) ` O ) ( O (Sect ` C ) O ) ( ( Id ` C ) ` O ) /\ ( ( Id ` C ) ` O ) ( O (Sect ` C ) O ) ( ( Id ` C ) ` O ) ) <-> ( ( ( ( Id ` C ) ` O ) ( <. O , O >. (comp ` C ) O ) ( ( Id ` C ) ` O ) ) = ( ( Id ` C ) ` O ) /\ ( ( ( Id ` C ) ` O ) ( <. O , O >. (comp ` C ) O ) ( ( Id ` C ) ` O ) ) = ( ( Id ` C ) ` O ) ) ) )
14	10, 10, 13	mpbir2and 957	. . . 4 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> ( ( ( Id ` C ) ` O ) ( O (Sect ` C ) O ) ( ( Id ` C ) ` O ) /\ ( ( Id ` C ) ` O ) ( O (Sect ` C ) O ) ( ( Id ` C ) ` O ) ) )
15	2, 5, 3, 4, 4, 11	isinv 16420	. . . 4 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> ( ( ( Id ` C ) ` O ) ( O (Inv ` C ) O ) ( ( Id ` C ) ` O ) <-> ( ( ( Id ` C ) ` O ) ( O (Sect ` C ) O ) ( ( Id ` C ) ` O ) /\ ( ( Id ` C ) ` O ) ( O (Sect ` C ) O ) ( ( Id ` C ) ` O ) ) ) )
16	14, 15	mpbird 247	. . 3 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> ( ( Id ` C ) ` O ) ( O (Inv ` C ) O ) ( ( Id ` C ) ` O ) )
17	2, 5, 3, 4, 4, 1, 16	inviso1 16426	. 2 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> ( ( Id ` C ) ` O ) e. ( O ( Iso ` C ) O ) )
18	1, 2, 3, 4, 4, 17	brcici 16460	1 |- ( ( C e. Cat /\ O e. ( Base ` C ) ) -> O ( ~=c𝑐 ` C ) O )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326  Sectcsect 16404  Invcinv 16405   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-rmo 2920  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-supp 7296  df-cat 16329  df-cid 16330  df-sect 16407  df-inv 16408  df-iso 16409  df-cic 16456

This theorem is referenced by:  cicer  16466