Theorem monpropd 16397

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
monpropd.3	|- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )
monpropd.4	|- ( ph -> (compf ` C ) = (compf ` D ) )
monpropd.c	|- ( ph -> C e. Cat )
monpropd.d	|- ( ph -> D e. Cat )

Assertion

Ref	Expression
monpropd	|- ( ph -> (Mono ` C ) = (Mono ` D ) )

Proof of Theorem monpropd

Dummy variables a b c f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . . . . . 12 |- ( Base ` C ) = ( Base ` C )
2		eqid 2622	. . . . . . . . . . . 12 |- ( Hom ` C ) = ( Hom ` C )
3		eqid 2622	. . . . . . . . . . . 12 |- ( Hom ` D ) = ( Hom ` D )
4		monpropd.3	. . . . . . . . . . . . . 14 |- ( ph -> ( Hom f ` C ) = ( Hom f ` D ) )
5	4	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) -> ( Hom f ` C ) = ( Hom f ` D ) )
6	5	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) -> ( Hom f ` C ) = ( Hom f ` D ) )
7		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) -> c e. ( Base ` C ) )
8		simp-4r 807	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) -> a e. ( Base ` C ) )
9	1, 2, 3, 6, 7, 8	homfeqval 16357	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) -> ( c ( Hom ` C ) a ) = ( c ( Hom ` D ) a ) )
10		eqid 2622	. . . . . . . . . . . 12 |- (comp ` C ) = (comp ` C )
11		eqid 2622	. . . . . . . . . . . 12 |- (comp ` D ) = (comp ` D )
12	4	ad5antr 770	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) /\ g e. ( c ( Hom ` C ) a ) ) -> ( Hom f ` C ) = ( Hom f ` D ) )
13		monpropd.4	. . . . . . . . . . . . 13 |- ( ph -> (compf ` C ) = (compf ` D ) )
14	13	ad5antr 770	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) /\ g e. ( c ( Hom ` C ) a ) ) -> (compf ` C ) = (compf ` D ) )
15		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) /\ g e. ( c ( Hom ` C ) a ) ) -> c e. ( Base ` C ) )
16		simp-5r 809	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) /\ g e. ( c ( Hom ` C ) a ) ) -> a e. ( Base ` C ) )
17		simp-4r 807	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) /\ g e. ( c ( Hom ` C ) a ) ) -> b e. ( Base ` C ) )
18		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) /\ g e. ( c ( Hom ` C ) a ) ) -> g e. ( c ( Hom ` C ) a ) )
19		simpllr 799	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) /\ g e. ( c ( Hom ` C ) a ) ) -> f e. ( a ( Hom ` C ) b ) )
20	1, 2, 10, 11, 12, 14, 15, 16, 17, 18, 19	comfeqval 16368	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) /\ g e. ( c ( Hom ` C ) a ) ) -> ( f ( <. c , a >. (comp ` C ) b ) g ) = ( f ( <. c , a >. (comp ` D ) b ) g ) )
21	9, 20	mpteq12dva 4732	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) -> ( g e. ( c ( Hom ` C ) a ) |-> ( f ( <. c , a >. (comp ` C ) b ) g ) ) = ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) )
22	21	cnveqd 5298	. . . . . . . . 9 |- ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) -> `' ( g e. ( c ( Hom ` C ) a ) |-> ( f ( <. c , a >. (comp ` C ) b ) g ) ) = `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) )
23	22	funeqd 5910	. . . . . . . 8 |- ( ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) /\ c e. ( Base ` C ) ) -> ( Fun `' ( g e. ( c ( Hom ` C ) a ) |-> ( f ( <. c , a >. (comp ` C ) b ) g ) ) <-> Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) ) )
24	23	ralbidva 2985	. . . . . . 7 |- ( ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) /\ f e. ( a ( Hom ` C ) b ) ) -> ( A. c e. ( Base ` C ) Fun `' ( g e. ( c ( Hom ` C ) a ) |-> ( f ( <. c , a >. (comp ` C ) b ) g ) ) <-> A. c e. ( Base ` C ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) ) )
25	24	rabbidva 3188	. . . . . 6 |- ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) -> { f e. ( a ( Hom ` C ) b ) | A. c e. ( Base ` C ) Fun `' ( g e. ( c ( Hom ` C ) a ) |-> ( f ( <. c , a >. (comp ` C ) b ) g ) ) } = { f e. ( a ( Hom ` C ) b ) | A. c e. ( Base ` C ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) } )
26		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) -> a e. ( Base ` C ) )
27		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) -> b e. ( Base ` C ) )
28	1, 2, 3, 5, 26, 27	homfeqval 16357	. . . . . . 7 |- ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) -> ( a ( Hom ` C ) b ) = ( a ( Hom ` D ) b ) )
29	4	homfeqbas 16356	. . . . . . . . 9 |- ( ph -> ( Base ` C ) = ( Base ` D ) )
30	29	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) -> ( Base ` C ) = ( Base ` D ) )
31	30	raleqdv 3144	. . . . . . 7 |- ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) -> ( A. c e. ( Base ` C ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) <-> A. c e. ( Base ` D ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) ) )
32	28, 31	rabeqbidv 3195	. . . . . 6 |- ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) -> { f e. ( a ( Hom ` C ) b ) | A. c e. ( Base ` C ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) } = { f e. ( a ( Hom ` D ) b ) | A. c e. ( Base ` D ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) } )
33	25, 32	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ a e. ( Base ` C ) ) /\ b e. ( Base ` C ) ) -> { f e. ( a ( Hom ` C ) b ) | A. c e. ( Base ` C ) Fun `' ( g e. ( c ( Hom ` C ) a ) |-> ( f ( <. c , a >. (comp ` C ) b ) g ) ) } = { f e. ( a ( Hom ` D ) b ) | A. c e. ( Base ` D ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) } )
34	33	3impa 1259	. . . 4 |- ( ( ph /\ a e. ( Base ` C ) /\ b e. ( Base ` C ) ) -> { f e. ( a ( Hom ` C ) b ) | A. c e. ( Base ` C ) Fun `' ( g e. ( c ( Hom ` C ) a ) |-> ( f ( <. c , a >. (comp ` C ) b ) g ) ) } = { f e. ( a ( Hom ` D ) b ) | A. c e. ( Base ` D ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) } )
35	34	mpt2eq3dva 6719	. . 3 |- ( ph -> ( a e. ( Base ` C ) , b e. ( Base ` C ) |-> { f e. ( a ( Hom ` C ) b ) | A. c e. ( Base ` C ) Fun `' ( g e. ( c ( Hom ` C ) a ) |-> ( f ( <. c , a >. (comp ` C ) b ) g ) ) } ) = ( a e. ( Base ` C ) , b e. ( Base ` C ) |-> { f e. ( a ( Hom ` D ) b ) | A. c e. ( Base ` D ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) } ) )
36		mpt2eq12 6715	. . . 4 |- ( ( ( Base ` C ) = ( Base ` D ) /\ ( Base ` C ) = ( Base ` D ) ) -> ( a e. ( Base ` C ) , b e. ( Base ` C ) |-> { f e. ( a ( Hom ` D ) b ) | A. c e. ( Base ` D ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) } ) = ( a e. ( Base ` D ) , b e. ( Base ` D ) |-> { f e. ( a ( Hom ` D ) b ) | A. c e. ( Base ` D ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) } ) )
37	29, 29, 36	syl2anc 693	. . 3 |- ( ph -> ( a e. ( Base ` C ) , b e. ( Base ` C ) |-> { f e. ( a ( Hom ` D ) b ) | A. c e. ( Base ` D ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) } ) = ( a e. ( Base ` D ) , b e. ( Base ` D ) |-> { f e. ( a ( Hom ` D ) b ) | A. c e. ( Base ` D ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) } ) )
38	35, 37	eqtrd 2656	. 2 |- ( ph -> ( a e. ( Base ` C ) , b e. ( Base ` C ) |-> { f e. ( a ( Hom ` C ) b ) | A. c e. ( Base ` C ) Fun `' ( g e. ( c ( Hom ` C ) a ) |-> ( f ( <. c , a >. (comp ` C ) b ) g ) ) } ) = ( a e. ( Base ` D ) , b e. ( Base ` D ) |-> { f e. ( a ( Hom ` D ) b ) | A. c e. ( Base ` D ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) } ) )
39		eqid 2622	. . 3 |- (Mono ` C ) = (Mono ` C )
40		monpropd.c	. . 3 |- ( ph -> C e. Cat )
41	1, 2, 10, 39, 40	monfval 16392	. 2 |- ( ph -> (Mono ` C ) = ( a e. ( Base ` C ) , b e. ( Base ` C ) |-> { f e. ( a ( Hom ` C ) b ) | A. c e. ( Base ` C ) Fun `' ( g e. ( c ( Hom ` C ) a ) |-> ( f ( <. c , a >. (comp ` C ) b ) g ) ) } ) )
42		eqid 2622	. . 3 |- ( Base ` D ) = ( Base ` D )
43		eqid 2622	. . 3 |- (Mono ` D ) = (Mono ` D )
44		monpropd.d	. . 3 |- ( ph -> D e. Cat )
45	42, 3, 11, 43, 44	monfval 16392	. 2 |- ( ph -> (Mono ` D ) = ( a e. ( Base ` D ) , b e. ( Base ` D ) |-> { f e. ( a ( Hom ` D ) b ) | A. c e. ( Base ` D ) Fun `' ( g e. ( c ( Hom ` D ) a ) |-> ( f ( <. c , a >. (comp ` D ) b ) g ) ) } ) )
46	38, 41, 45	3eqtr4d 2666	1 |- ( ph -> (Mono ` C ) = (Mono ` D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   <.cop 4183   |-> cmpt 4729   `'ccnv 5113   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Hom _f chomf 16327  comp_fccomf 16328  Monocmon 16388

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  df-comf 16332  df-mon 16390

This theorem is referenced by:  oppcepi  16399