Theorem cicer 16466

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.

Step	Hyp	Ref	Expression
1		relopab 5247	. . . . . 6 |- Rel { <. x , y >. | ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ ( ( Iso ` C ) ` <. x , y >. ) =/= (/) ) }
2	1	a1i 11	. . . . 5 |- ( C e. Cat -> Rel { <. x , y >. | ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ ( ( Iso ` C ) ` <. x , y >. ) =/= (/) ) } )
3		fveq2 6191	. . . . . . . . 9 |- ( f = <. x , y >. -> ( ( Iso ` C ) ` f ) = ( ( Iso ` C ) ` <. x , y >. ) )
4	3	neeq1d 2853	. . . . . . . 8 |- ( f = <. x , y >. -> ( ( ( Iso ` C ) ` f ) =/= (/) <-> ( ( Iso ` C ) ` <. x , y >. ) =/= (/) ) )
5	4	rabxp 5154	. . . . . . 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 >. ) =/= (/) ) }
6	5	a1i 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 >. ) =/= (/) ) } )
7	6	releqd 5203	. . . . 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 >. ) =/= (/) ) } ) )
8	2, 7	mpbird 247	. . . 4 |- ( C e. Cat -> Rel { f e. ( ( Base ` C ) X. ( Base ` C ) ) | ( ( Iso ` C ) ` f ) =/= (/) } )
9		isofn 16435	. . . . . 6 |- ( C e. Cat -> ( Iso ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
10		fvex 6201	. . . . . . 7 |- ( Base ` C ) e. _V
11		sqxpexg 6963	. . . . . . 7 |- ( ( Base ` C ) e. _V -> ( ( Base ` C ) X. ( Base ` C ) ) e. _V )
12	10, 11	mp1i 13	. . . . . 6 |- ( C e. Cat -> ( ( Base ` C ) X. ( Base ` C ) ) e. _V )
13		0ex 4790	. . . . . . 7 |- (/) e. _V
14	13	a1i 11	. . . . . 6 |- ( C e. Cat -> (/) e. _V )
15		suppvalfn 7302	. . . . . 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 ) =/= (/) } )
16	9, 12, 14, 15	syl3anc 1326	. . . . 5 |- ( C e. Cat -> ( ( Iso ` C ) supp (/) ) = { f e. ( ( Base ` C ) X. ( Base ` C ) ) | ( ( Iso ` C ) ` f ) =/= (/) } )
17	16	releqd 5203	. . . 4 |- ( C e. Cat -> ( Rel ( ( Iso ` C ) supp (/) ) <-> Rel { f e. ( ( Base ` C ) X. ( Base ` C ) ) | ( ( Iso ` C ) ` f ) =/= (/) } ) )
18	8, 17	mpbird 247	. . 3 |- ( C e. Cat -> Rel ( ( Iso ` C ) supp (/) ) )
19		cicfval 16457	. . . 4 |- ( C e. Cat -> ( ~=c𝑐 ` C ) = ( ( Iso ` C ) supp (/) ) )
20	19	releqd 5203	. . 3 |- ( C e. Cat -> ( Rel ( ~=c𝑐 ` C ) <-> Rel ( ( Iso ` C ) supp (/) ) ) )
21	18, 20	mpbird 247	. 2 |- ( C e. Cat -> Rel ( ~=c𝑐 ` C ) )
22		cicsym 16464	. 2 |- ( ( C e. Cat /\ x ( ~=c𝑐 ` C ) y ) -> y ( ~=c𝑐 ` C ) x )
23		cictr 16465	. . 3 |- ( ( C e. Cat /\ x ( ~=c𝑐 ` C ) y /\ y ( ~=c𝑐 ` C ) z ) -> x ( ~=c𝑐 ` C ) z )
24	23	3expb 1266	. 2 |- ( ( C e. Cat /\ ( x ( ~=c𝑐 ` C ) y /\ y ( ~=c𝑐 ` C ) z ) ) -> x ( ~=c𝑐 ` C ) z )
25		cicref 16461	. . 3 |- ( ( C e. Cat /\ x e. ( Base ` C ) ) -> x ( ~=c𝑐 ` C ) x )
26		ciclcl 16462	. . 3 |- ( ( C e. Cat /\ x ( ~=c𝑐 ` C ) x ) -> x e. ( Base ` C ) )
27	25, 26	impbida 877	. 2 |- ( C e. Cat -> ( x e. ( Base ` C ) <-> x ( ~=c𝑐 ` C ) x ) )
28	21, 22, 24, 27	iserd 7768	1 |- ( C e. Cat -> ( ~=c𝑐 ` C ) Er ( Base ` C ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   (/)c0 3915   <.cop 4183   class class class wbr 4653   {copab 4712   X. cxp 5112   Rel wrel 5119   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Er wer 7739   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-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-er 7742  df-cat 16329  df-cid 16330  df-sect 16407  df-inv 16408  df-iso 16409  df-cic 16456

This theorem is referenced by: (None)