Theorem fthmon 16587

Description: A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
fthmon.b	|- B = ( Base ` C )
fthmon.h	|- H = ( Hom ` C )
fthmon.f	|- ( ph -> F ( C Faith D ) G )
fthmon.x	|- ( ph -> X e. B )
fthmon.y	|- ( ph -> Y e. B )
fthmon.r	|- ( ph -> R e. ( X H Y ) )
fthmon.m	|- M = (Mono ` C )
fthmon.n	|- N = (Mono ` D )
fthmon.1	|- ( ph -> ( ( X G Y ) ` R ) e. ( ( F ` X ) N ( F ` Y ) ) )

Assertion

Ref	Expression
fthmon	|- ( ph -> R e. ( X M Y ) )

Proof of Theorem fthmon

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

Step	Hyp	Ref	Expression
1		fthmon.r	. 2 |- ( ph -> R e. ( X H Y ) )
2		eqid 2622	. . . . . 6 |- ( Base ` D ) = ( Base ` D )
3		eqid 2622	. . . . . 6 |- ( Hom ` D ) = ( Hom ` D )
4		eqid 2622	. . . . . 6 |- (comp ` D ) = (comp ` D )
5		fthmon.n	. . . . . 6 |- N = (Mono ` D )
6		fthmon.f	. . . . . . . . . . 11 |- ( ph -> F ( C Faith D ) G )
7		fthfunc 16567	. . . . . . . . . . . 12 |- ( C Faith D ) C_ ( C Func D )
8	7	ssbri 4697	. . . . . . . . . . 11 |- ( F ( C Faith D ) G -> F ( C Func D ) G )
9	6, 8	syl 17	. . . . . . . . . 10 |- ( ph -> F ( C Func D ) G )
10		df-br 4654	. . . . . . . . . 10 |- ( F ( C Func D ) G <-> <. F , G >. e. ( C Func D ) )
11	9, 10	sylib 208	. . . . . . . . 9 |- ( ph -> <. F , G >. e. ( C Func D ) )
12		funcrcl 16523	. . . . . . . . 9 |- ( <. F , G >. e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
13	11, 12	syl 17	. . . . . . . 8 |- ( ph -> ( C e. Cat /\ D e. Cat ) )
14	13	simprd 479	. . . . . . 7 |- ( ph -> D e. Cat )
15	14	adantr 481	. . . . . 6 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> D e. Cat )
16		fthmon.b	. . . . . . . 8 |- B = ( Base ` C )
17	9	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> F ( C Func D ) G )
18	16, 2, 17	funcf1 16526	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> F : B --> ( Base ` D ) )
19		fthmon.x	. . . . . . . 8 |- ( ph -> X e. B )
20	19	adantr 481	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> X e. B )
21	18, 20	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( F ` X ) e. ( Base ` D ) )
22		fthmon.y	. . . . . . . 8 |- ( ph -> Y e. B )
23	22	adantr 481	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> Y e. B )
24	18, 23	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( F ` Y ) e. ( Base ` D ) )
25		simpr1 1067	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> z e. B )
26	18, 25	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( F ` z ) e. ( Base ` D ) )
27		fthmon.1	. . . . . . 7 |- ( ph -> ( ( X G Y ) ` R ) e. ( ( F ` X ) N ( F ` Y ) ) )
28	27	adantr 481	. . . . . 6 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( ( X G Y ) ` R ) e. ( ( F ` X ) N ( F ` Y ) ) )
29		fthmon.h	. . . . . . . 8 |- H = ( Hom ` C )
30	16, 29, 3, 17, 25, 20	funcf2 16528	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( z G X ) : ( z H X ) --> ( ( F ` z ) ( Hom ` D ) ( F ` X ) ) )
31		simpr2 1068	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> f e. ( z H X ) )
32	30, 31	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( ( z G X ) ` f ) e. ( ( F ` z ) ( Hom ` D ) ( F ` X ) ) )
33		simpr3 1069	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> g e. ( z H X ) )
34	30, 33	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( ( z G X ) ` g ) e. ( ( F ` z ) ( Hom ` D ) ( F ` X ) ) )
35	2, 3, 4, 5, 15, 21, 24, 26, 28, 32, 34	moni 16396	. . . . 5 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( ( ( ( X G Y ) ` R ) ( <. ( F ` z ) , ( F ` X ) >. (comp ` D ) ( F ` Y ) ) ( ( z G X ) ` f ) ) = ( ( ( X G Y ) ` R ) ( <. ( F ` z ) , ( F ` X ) >. (comp ` D ) ( F ` Y ) ) ( ( z G X ) ` g ) ) <-> ( ( z G X ) ` f ) = ( ( z G X ) ` g ) ) )
36		eqid 2622	. . . . . . . 8 |- (comp ` C ) = (comp ` C )
37	1	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> R e. ( X H Y ) )
38	16, 29, 36, 4, 17, 25, 20, 23, 31, 37	funcco 16531	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( ( z G Y ) ` ( R ( <. z , X >. (comp ` C ) Y ) f ) ) = ( ( ( X G Y ) ` R ) ( <. ( F ` z ) , ( F ` X ) >. (comp ` D ) ( F ` Y ) ) ( ( z G X ) ` f ) ) )
39	16, 29, 36, 4, 17, 25, 20, 23, 33, 37	funcco 16531	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( ( z G Y ) ` ( R ( <. z , X >. (comp ` C ) Y ) g ) ) = ( ( ( X G Y ) ` R ) ( <. ( F ` z ) , ( F ` X ) >. (comp ` D ) ( F ` Y ) ) ( ( z G X ) ` g ) ) )
40	38, 39	eqeq12d 2637	. . . . . 6 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( ( ( z G Y ) ` ( R ( <. z , X >. (comp ` C ) Y ) f ) ) = ( ( z G Y ) ` ( R ( <. z , X >. (comp ` C ) Y ) g ) ) <-> ( ( ( X G Y ) ` R ) ( <. ( F ` z ) , ( F ` X ) >. (comp ` D ) ( F ` Y ) ) ( ( z G X ) ` f ) ) = ( ( ( X G Y ) ` R ) ( <. ( F ` z ) , ( F ` X ) >. (comp ` D ) ( F ` Y ) ) ( ( z G X ) ` g ) ) ) )
41	6	adantr 481	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> F ( C Faith D ) G )
42	13	simpld 475	. . . . . . . . 9 |- ( ph -> C e. Cat )
43	42	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> C e. Cat )
44	16, 29, 36, 43, 25, 20, 23, 31, 37	catcocl 16346	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( R ( <. z , X >. (comp ` C ) Y ) f ) e. ( z H Y ) )
45	16, 29, 36, 43, 25, 20, 23, 33, 37	catcocl 16346	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( R ( <. z , X >. (comp ` C ) Y ) g ) e. ( z H Y ) )
46	16, 29, 3, 41, 25, 23, 44, 45	fthi 16578	. . . . . 6 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( ( ( z G Y ) ` ( R ( <. z , X >. (comp ` C ) Y ) f ) ) = ( ( z G Y ) ` ( R ( <. z , X >. (comp ` C ) Y ) g ) ) <-> ( R ( <. z , X >. (comp ` C ) Y ) f ) = ( R ( <. z , X >. (comp ` C ) Y ) g ) ) )
47	40, 46	bitr3d 270	. . . . 5 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( ( ( ( X G Y ) ` R ) ( <. ( F ` z ) , ( F ` X ) >. (comp ` D ) ( F ` Y ) ) ( ( z G X ) ` f ) ) = ( ( ( X G Y ) ` R ) ( <. ( F ` z ) , ( F ` X ) >. (comp ` D ) ( F ` Y ) ) ( ( z G X ) ` g ) ) <-> ( R ( <. z , X >. (comp ` C ) Y ) f ) = ( R ( <. z , X >. (comp ` C ) Y ) g ) ) )
48	16, 29, 3, 41, 25, 20, 31, 33	fthi 16578	. . . . 5 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( ( ( z G X ) ` f ) = ( ( z G X ) ` g ) <-> f = g ) )
49	35, 47, 48	3bitr3d 298	. . . 4 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( ( R ( <. z , X >. (comp ` C ) Y ) f ) = ( R ( <. z , X >. (comp ` C ) Y ) g ) <-> f = g ) )
50	49	biimpd 219	. . 3 |- ( ( ph /\ ( z e. B /\ f e. ( z H X ) /\ g e. ( z H X ) ) ) -> ( ( R ( <. z , X >. (comp ` C ) Y ) f ) = ( R ( <. z , X >. (comp ` C ) Y ) g ) -> f = g ) )
51	50	ralrimivvva 2972	. 2 |- ( ph -> A. z e. B A. f e. ( z H X ) A. g e. ( z H X ) ( ( R ( <. z , X >. (comp ` C ) Y ) f ) = ( R ( <. z , X >. (comp ` C ) Y ) g ) -> f = g ) )
52		fthmon.m	. . 3 |- M = (Mono ` C )
53	16, 29, 36, 52, 42, 19, 22	ismon2 16394	. 2 |- ( ph -> ( R e. ( X M Y ) <-> ( R e. ( X H Y ) /\ A. z e. B A. f e. ( z H X ) A. g e. ( z H X ) ( ( R ( <. z , X >. (comp ` C ) Y ) f ) = ( R ( <. z , X >. (comp ` C ) Y ) g ) -> f = g ) ) ) )
54	1, 51, 53	mpbir2and 957	1 |- ( ph -> R e. ( X M Y ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325  Monocmon 16388   Func cfunc 16514   Faith cfth 16563

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-map 7859  df-ixp 7909  df-cat 16329  df-mon 16390  df-func 16518  df-fth 16565

This theorem is referenced by:  fthepi  16588