Theorem isofn 16435

Description: The function value of the function returning the isomorphisms of a category is a function over the square product of the base set of the category. (Contributed by AV, 5-Apr-2017.)

Assertion

Ref	Expression
isofn	|- ( C e. Cat -> ( Iso ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )

Proof of Theorem isofn

Dummy variables c x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmexg 7097	. . . . . 6 |- ( x e. _V -> dom x e. _V )
2	1	adantl 482	. . . . 5 |- ( ( C e. Cat /\ x e. _V ) -> dom x e. _V )
3	2	ralrimiva 2966	. . . 4 |- ( C e. Cat -> A. x e. _V dom x e. _V )
4		eqid 2622	. . . . 5 |- ( x e. _V |-> dom x ) = ( x e. _V |-> dom x )
5	4	fnmpt 6020	. . . 4 |- ( A. x e. _V dom x e. _V -> ( x e. _V |-> dom x ) Fn _V )
6	3, 5	syl 17	. . 3 |- ( C e. Cat -> ( x e. _V |-> dom x ) Fn _V )
7		ovex 6678	. . . . . . . 8 |- ( x (Sect ` C ) y ) e. _V
8	7	inex1 4799	. . . . . . 7 |- ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) e. _V
9	8	a1i 11	. . . . . 6 |- ( ( C e. Cat /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) e. _V )
10	9	ralrimivva 2971	. . . . 5 |- ( C e. Cat -> A. x e. ( Base ` C ) A. y e. ( Base ` C ) ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) e. _V )
11		eqid 2622	. . . . . 6 |- ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) )
12	11	fnmpt2 7238	. . . . 5 |- ( A. x e. ( Base ` C ) A. y e. ( Base ` C ) ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) e. _V -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
13	10, 12	syl 17	. . . 4 |- ( C e. Cat -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
14		df-inv 16408	. . . . . . 7 |- Inv = ( c e. Cat |-> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> ( ( x (Sect ` c ) y ) i^i `' ( y (Sect ` c ) x ) ) ) )
15	14	a1i 11	. . . . . 6 |- ( C e. Cat -> Inv = ( c e. Cat |-> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> ( ( x (Sect ` c ) y ) i^i `' ( y (Sect ` c ) x ) ) ) ) )
16		fveq2 6191	. . . . . . . 8 |- ( c = C -> ( Base ` c ) = ( Base ` C ) )
17		fveq2 6191	. . . . . . . . . 10 |- ( c = C -> (Sect ` c ) = (Sect ` C ) )
18	17	oveqd 6667	. . . . . . . . 9 |- ( c = C -> ( x (Sect ` c ) y ) = ( x (Sect ` C ) y ) )
19	17	oveqd 6667	. . . . . . . . . 10 |- ( c = C -> ( y (Sect ` c ) x ) = ( y (Sect ` C ) x ) )
20	19	cnveqd 5298	. . . . . . . . 9 |- ( c = C -> `' ( y (Sect ` c ) x ) = `' ( y (Sect ` C ) x ) )
21	18, 20	ineq12d 3815	. . . . . . . 8 |- ( c = C -> ( ( x (Sect ` c ) y ) i^i `' ( y (Sect ` c ) x ) ) = ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) )
22	16, 16, 21	mpt2eq123dv 6717	. . . . . . 7 |- ( c = C -> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> ( ( x (Sect ` c ) y ) i^i `' ( y (Sect ` c ) x ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) ) )
23	22	adantl 482	. . . . . 6 |- ( ( C e. Cat /\ c = C ) -> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> ( ( x (Sect ` c ) y ) i^i `' ( y (Sect ` c ) x ) ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) ) )
24		id 22	. . . . . 6 |- ( C e. Cat -> C e. Cat )
25		fvex 6201	. . . . . . . 8 |- ( Base ` C ) e. _V
26	25, 25	pm3.2i 471	. . . . . . 7 |- ( ( Base ` C ) e. _V /\ ( Base ` C ) e. _V )
27	11	mpt2exg 7245	. . . . . . 7 |- ( ( ( Base ` C ) e. _V /\ ( Base ` C ) e. _V ) -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) ) e. _V )
28	26, 27	mp1i 13	. . . . . 6 |- ( C e. Cat -> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) ) e. _V )
29	15, 23, 24, 28	fvmptd 6288	. . . . 5 |- ( C e. Cat -> (Inv ` C ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) ) )
30	29	fneq1d 5981	. . . 4 |- ( C e. Cat -> ( (Inv ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) <-> ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( ( x (Sect ` C ) y ) i^i `' ( y (Sect ` C ) x ) ) ) Fn ( ( Base ` C ) X. ( Base ` C ) ) ) )
31	13, 30	mpbird 247	. . 3 |- ( C e. Cat -> (Inv ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
32		ssv 3625	. . . 4 |- ran (Inv ` C ) C_ _V
33	32	a1i 11	. . 3 |- ( C e. Cat -> ran (Inv ` C ) C_ _V )
34		fnco 5999	. . 3 |- ( ( ( x e. _V |-> dom x ) Fn _V /\ (Inv ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) /\ ran (Inv ` C ) C_ _V ) -> ( ( x e. _V |-> dom x ) o. (Inv ` C ) ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
35	6, 31, 33, 34	syl3anc 1326	. 2 |- ( C e. Cat -> ( ( x e. _V |-> dom x ) o. (Inv ` C ) ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )
36		isofval 16417	. . 3 |- ( C e. Cat -> ( Iso ` C ) = ( ( x e. _V |-> dom x ) o. (Inv ` C ) ) )
37	36	fneq1d 5981	. 2 |- ( C e. Cat -> ( ( Iso ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) <-> ( ( x e. _V |-> dom x ) o. (Inv ` C ) ) Fn ( ( Base ` C ) X. ( Base ` C ) ) ) )
38	35, 37	mpbird 247	1 |- ( C e. Cat -> ( Iso ` C ) Fn ( ( Base ` C ) X. ( Base ` C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   o. ccom 5118   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   Catccat 16325  Sectcsect 16404  Invcinv 16405   Iso ciso 16406

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-inv 16408  df-iso 16409

This theorem is referenced by:  brcic  16458  ciclcl  16462  cicrcl  16463  cicer  16466