Theorem fullpropd 16580

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same full functors. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
fullpropd.1	|- ( ph -> ( Hom f ` A ) = ( Hom f ` B ) )
fullpropd.2	|- ( ph -> (compf ` A ) = (compf ` B ) )
fullpropd.3	|- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )
fullpropd.4	|- ( ph -> (compf ` C ) = (compf ` D ) )
fullpropd.a	|- ( ph -> A e. V )
fullpropd.b	|- ( ph -> B e. V )
fullpropd.c	|- ( ph -> C e. V )
fullpropd.d	|- ( ph -> D e. V )

Assertion

Ref	Expression
fullpropd	|- ( ph -> ( A Full C ) = ( B Full D ) )

Proof of Theorem fullpropd

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

Step	Hyp	Ref	Expression
1		relfull 16568	. 2 |- Rel ( A Full C )
2		relfull 16568	. 2 |- Rel ( B Full D )
3		fullpropd.1	. . . . . . . 8 |- ( ph -> ( Hom f ` A ) = ( Hom f ` B ) )
4	3	homfeqbas 16356	. . . . . . 7 |- ( ph -> ( Base ` A ) = ( Base ` B ) )
5	4	adantr 481	. . . . . 6 |- ( ( ph /\ f ( A Func C ) g ) -> ( Base ` A ) = ( Base ` B ) )
6	5	adantr 481	. . . . . . 7 |- ( ( ( ph /\ f ( A Func C ) g ) /\ x e. ( Base ` A ) ) -> ( Base ` A ) = ( Base ` B ) )
7		eqid 2622	. . . . . . . . 9 |- ( Base ` C ) = ( Base ` C )
8		eqid 2622	. . . . . . . . 9 |- ( Hom ` C ) = ( Hom ` C )
9		eqid 2622	. . . . . . . . 9 |- ( Hom ` D ) = ( Hom ` D )
10		fullpropd.3	. . . . . . . . . 10 |- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )
11	10	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ f ( A Func C ) g ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> ( Hom f ` C ) = ( Hom f ` D ) )
12		eqid 2622	. . . . . . . . . . 11 |- ( Base ` A ) = ( Base ` A )
13		simpllr 799	. . . . . . . . . . 11 |- ( ( ( ( ph /\ f ( A Func C ) g ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> f ( A Func C ) g )
14	12, 7, 13	funcf1 16526	. . . . . . . . . 10 |- ( ( ( ( ph /\ f ( A Func C ) g ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> f : ( Base ` A ) --> ( Base ` C ) )
15		simplr 792	. . . . . . . . . 10 |- ( ( ( ( ph /\ f ( A Func C ) g ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> x e. ( Base ` A ) )
16	14, 15	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ( ( ph /\ f ( A Func C ) g ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> ( f ` x ) e. ( Base ` C ) )
17		simpr 477	. . . . . . . . . 10 |- ( ( ( ( ph /\ f ( A Func C ) g ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> y e. ( Base ` A ) )
18	14, 17	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ( ( ph /\ f ( A Func C ) g ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> ( f ` y ) e. ( Base ` C ) )
19	7, 8, 9, 11, 16, 18	homfeqval 16357	. . . . . . . 8 |- ( ( ( ( ph /\ f ( A Func C ) g ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> ( ( f ` x ) ( Hom ` C ) ( f ` y ) ) = ( ( f ` x ) ( Hom ` D ) ( f ` y ) ) )
20	19	eqeq2d 2632	. . . . . . 7 |- ( ( ( ( ph /\ f ( A Func C ) g ) /\ x e. ( Base ` A ) ) /\ y e. ( Base ` A ) ) -> ( ran ( x g y ) = ( ( f ` x ) ( Hom ` C ) ( f ` y ) ) <-> ran ( x g y ) = ( ( f ` x ) ( Hom ` D ) ( f ` y ) ) ) )
21	6, 20	raleqbidva 3154	. . . . . 6 |- ( ( ( ph /\ f ( A Func C ) g ) /\ x e. ( Base ` A ) ) -> ( A. y e. ( Base ` A ) ran ( x g y ) = ( ( f ` x ) ( Hom ` C ) ( f ` y ) ) <-> A. y e. ( Base ` B ) ran ( x g y ) = ( ( f ` x ) ( Hom ` D ) ( f ` y ) ) ) )
22	5, 21	raleqbidva 3154	. . . . 5 |- ( ( ph /\ f ( A Func C ) g ) -> ( A. x e. ( Base ` A ) A. y e. ( Base ` A ) ran ( x g y ) = ( ( f ` x ) ( Hom ` C ) ( f ` y ) ) <-> A. x e. ( Base ` B ) A. y e. ( Base ` B ) ran ( x g y ) = ( ( f ` x ) ( Hom ` D ) ( f ` y ) ) ) )
23	22	pm5.32da 673	. . . 4 |- ( ph -> ( ( f ( A Func C ) g /\ A. x e. ( Base ` A ) A. y e. ( Base ` A ) ran ( x g y ) = ( ( f ` x ) ( Hom ` C ) ( f ` y ) ) ) <-> ( f ( A Func C ) g /\ A. x e. ( Base ` B ) A. y e. ( Base ` B ) ran ( x g y ) = ( ( f ` x ) ( Hom ` D ) ( f ` y ) ) ) ) )
24		fullpropd.2	. . . . . . 7 |- ( ph -> (compf ` A ) = (compf ` B ) )
25		fullpropd.4	. . . . . . 7 |- ( ph -> (compf ` C ) = (compf ` D ) )
26		fullpropd.a	. . . . . . 7 |- ( ph -> A e. V )
27		fullpropd.b	. . . . . . 7 |- ( ph -> B e. V )
28		fullpropd.c	. . . . . . 7 |- ( ph -> C e. V )
29		fullpropd.d	. . . . . . 7 |- ( ph -> D e. V )
30	3, 24, 10, 25, 26, 27, 28, 29	funcpropd 16560	. . . . . 6 |- ( ph -> ( A Func C ) = ( B Func D ) )
31	30	breqd 4664	. . . . 5 |- ( ph -> ( f ( A Func C ) g <-> f ( B Func D ) g ) )
32	31	anbi1d 741	. . . 4 |- ( ph -> ( ( f ( A Func C ) g /\ A. x e. ( Base ` B ) A. y e. ( Base ` B ) ran ( x g y ) = ( ( f ` x ) ( Hom ` D ) ( f ` y ) ) ) <-> ( f ( B Func D ) g /\ A. x e. ( Base ` B ) A. y e. ( Base ` B ) ran ( x g y ) = ( ( f ` x ) ( Hom ` D ) ( f ` y ) ) ) ) )
33	23, 32	bitrd 268	. . 3 |- ( ph -> ( ( f ( A Func C ) g /\ A. x e. ( Base ` A ) A. y e. ( Base ` A ) ran ( x g y ) = ( ( f ` x ) ( Hom ` C ) ( f ` y ) ) ) <-> ( f ( B Func D ) g /\ A. x e. ( Base ` B ) A. y e. ( Base ` B ) ran ( x g y ) = ( ( f ` x ) ( Hom ` D ) ( f ` y ) ) ) ) )
34	12, 8	isfull 16570	. . 3 |- ( f ( A Full C ) g <-> ( f ( A Func C ) g /\ A. x e. ( Base ` A ) A. y e. ( Base ` A ) ran ( x g y ) = ( ( f ` x ) ( Hom ` C ) ( f ` y ) ) ) )
35		eqid 2622	. . . 4 |- ( Base ` B ) = ( Base ` B )
36	35, 9	isfull 16570	. . 3 |- ( f ( B Full D ) g <-> ( f ( B Func D ) g /\ A. x e. ( Base ` B ) A. y e. ( Base ` B ) ran ( x g y ) = ( ( f ` x ) ( Hom ` D ) ( f ` y ) ) ) )
37	33, 34, 36	3bitr4g 303	. 2 |- ( ph -> ( f ( A Full C ) g <-> f ( B Full D ) g ) )
38	1, 2, 37	eqbrrdiv 5218	1 |- ( ph -> ( A Full C ) = ( B Full D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   ran crn 5115   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952   Hom _f chomf 16327  comp_fccomf 16328   Func cfunc 16514   Full cful 16562

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-fal 1489  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-cat 16329  df-cid 16330  df-homf 16331  df-comf 16332  df-func 16518  df-full 16564

This theorem is referenced by: (None)