Theorem homfeq 16354

Description: Condition for two categories with the same base to have the same hom-sets. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
homfeq.h	|- H = ( Hom ` C )
homfeq.j	|- J = ( Hom ` D )
homfeq.1	|- ( ph -> B = ( Base ` C ) )
homfeq.2	|- ( ph -> B = ( Base ` D ) )

Assertion

Ref	Expression
homfeq	|- ( ph -> ( ( Hom f ` C ) = ( Hom f ` D ) <-> A. x e. B A. y e. B ( x H y ) = ( x J y ) ) )

Distinct variable groups:   x, y, B   x, C, y   x, D, y   x, H, y   ph, x, y   x, J, y

Proof of Theorem homfeq

Step	Hyp	Ref	Expression
1		homfeq.1	. . . . 5 |- ( ph -> B = ( Base ` C ) )
2		eqidd 2623	. . . . 5 |- ( ph -> ( x H y ) = ( x H y ) )
3	1, 1, 2	mpt2eq123dv 6717	. . . 4 |- ( ph -> ( x e. B , y e. B |-> ( x H y ) ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x H y ) ) )
4		eqid 2622	. . . . 5 |- ( Hom f ` C ) = ( Hom f ` C )
5		eqid 2622	. . . . 5 |- ( Base ` C ) = ( Base ` C )
6		homfeq.h	. . . . 5 |- H = ( Hom ` C )
7	4, 5, 6	homffval 16350	. . . 4 |- ( Hom f ` C ) = ( x e. ( Base ` C ) , y e. ( Base ` C ) |-> ( x H y ) )
8	3, 7	syl6reqr 2675	. . 3 |- ( ph -> ( Hom f ` C ) = ( x e. B , y e. B |-> ( x H y ) ) )
9		homfeq.2	. . . . 5 |- ( ph -> B = ( Base ` D ) )
10		eqidd 2623	. . . . 5 |- ( ph -> ( x J y ) = ( x J y ) )
11	9, 9, 10	mpt2eq123dv 6717	. . . 4 |- ( ph -> ( x e. B , y e. B |-> ( x J y ) ) = ( x e. ( Base ` D ) , y e. ( Base ` D ) |-> ( x J y ) ) )
12		eqid 2622	. . . . 5 |- ( Hom f ` D ) = ( Hom f ` D )
13		eqid 2622	. . . . 5 |- ( Base ` D ) = ( Base ` D )
14		homfeq.j	. . . . 5 |- J = ( Hom ` D )
15	12, 13, 14	homffval 16350	. . . 4 |- ( Hom f ` D ) = ( x e. ( Base ` D ) , y e. ( Base ` D ) |-> ( x J y ) )
16	11, 15	syl6reqr 2675	. . 3 |- ( ph -> ( Hom f ` D ) = ( x e. B , y e. B |-> ( x J y ) ) )
17	8, 16	eqeq12d 2637	. 2 |- ( ph -> ( ( Hom f ` C ) = ( Hom f ` D ) <-> ( x e. B , y e. B |-> ( x H y ) ) = ( x e. B , y e. B |-> ( x J y ) ) ) )
18		ovex 6678	. . . 4 |- ( x H y ) e. _V
19	18	rgen2w 2925	. . 3 |- A. x e. B A. y e. B ( x H y ) e. _V
20		mpt22eqb 6769	. . 3 |- ( A. x e. B A. y e. B ( x H y ) e. _V -> ( ( x e. B , y e. B |-> ( x H y ) ) = ( x e. B , y e. B |-> ( x J y ) ) <-> A. x e. B A. y e. B ( x H y ) = ( x J y ) ) )
21	19, 20	ax-mp 5	. 2 |- ( ( x e. B , y e. B |-> ( x H y ) ) = ( x e. B , y e. B |-> ( x J y ) ) <-> A. x e. B A. y e. B ( x H y ) = ( x J y ) )
22	17, 21	syl6bb 276	1 |- ( ph -> ( ( Hom f ` C ) = ( Hom f ` D ) <-> A. x e. B A. y e. B ( x H y ) = ( x J y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   Hom chom 15952   Hom _f chomf 16327

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-homf 16331

This theorem is referenced by:  homfeqd  16355  fullresc  16511  resssetc  16742  resscatc  16755