Theorem moni 16396

Description: Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
ismon.b	|- B = ( Base ` C )
ismon.h	|- H = ( Hom ` C )
ismon.o	|- .x. = (comp ` C )
ismon.s	|- M = (Mono ` C )
ismon.c	|- ( ph -> C e. Cat )
ismon.x	|- ( ph -> X e. B )
ismon.y	|- ( ph -> Y e. B )
moni.z	|- ( ph -> Z e. B )
moni.f	|- ( ph -> F e. ( X M Y ) )
moni.g	|- ( ph -> G e. ( Z H X ) )
moni.k	|- ( ph -> K e. ( Z H X ) )

Assertion

Ref	Expression
moni	|- ( ph -> ( ( F ( <. Z , X >. .x. Y ) G ) = ( F ( <. Z , X >. .x. Y ) K ) <-> G = K ) )

Proof of Theorem moni

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

Step	Hyp	Ref	Expression
1		moni.f	. . . . 5 |- ( ph -> F e. ( X M Y ) )
2		ismon.b	. . . . . 6 |- B = ( Base ` C )
3		ismon.h	. . . . . 6 |- H = ( Hom ` C )
4		ismon.o	. . . . . 6 |- .x. = (comp ` C )
5		ismon.s	. . . . . 6 |- M = (Mono ` C )
6		ismon.c	. . . . . 6 |- ( ph -> C e. Cat )
7		ismon.x	. . . . . 6 |- ( ph -> X e. B )
8		ismon.y	. . . . . 6 |- ( ph -> Y e. B )
9	2, 3, 4, 5, 6, 7, 8	ismon2 16394	. . . . 5 |- ( ph -> ( F e. ( X M Y ) <-> ( F e. ( X H Y ) /\ A. z e. B A. g e. ( z H X ) A. h e. ( z H X ) ( ( F ( <. z , X >. .x. Y ) g ) = ( F ( <. z , X >. .x. Y ) h ) -> g = h ) ) ) )
10	1, 9	mpbid 222	. . . 4 |- ( ph -> ( F e. ( X H Y ) /\ A. z e. B A. g e. ( z H X ) A. h e. ( z H X ) ( ( F ( <. z , X >. .x. Y ) g ) = ( F ( <. z , X >. .x. Y ) h ) -> g = h ) ) )
11	10	simprd 479	. . 3 |- ( ph -> A. z e. B A. g e. ( z H X ) A. h e. ( z H X ) ( ( F ( <. z , X >. .x. Y ) g ) = ( F ( <. z , X >. .x. Y ) h ) -> g = h ) )
12		moni.z	. . . 4 |- ( ph -> Z e. B )
13		moni.g	. . . . . . 7 |- ( ph -> G e. ( Z H X ) )
14	13	adantr 481	. . . . . 6 |- ( ( ph /\ z = Z ) -> G e. ( Z H X ) )
15		simpr 477	. . . . . . 7 |- ( ( ph /\ z = Z ) -> z = Z )
16	15	oveq1d 6665	. . . . . 6 |- ( ( ph /\ z = Z ) -> ( z H X ) = ( Z H X ) )
17	14, 16	eleqtrrd 2704	. . . . 5 |- ( ( ph /\ z = Z ) -> G e. ( z H X ) )
18		moni.k	. . . . . . . . 9 |- ( ph -> K e. ( Z H X ) )
19	18	adantr 481	. . . . . . . 8 |- ( ( ph /\ z = Z ) -> K e. ( Z H X ) )
20	19, 16	eleqtrrd 2704	. . . . . . 7 |- ( ( ph /\ z = Z ) -> K e. ( z H X ) )
21	20	adantr 481	. . . . . 6 |- ( ( ( ph /\ z = Z ) /\ g = G ) -> K e. ( z H X ) )
22		simpllr 799	. . . . . . . . . . 11 |- ( ( ( ( ph /\ z = Z ) /\ g = G ) /\ h = K ) -> z = Z )
23	22	opeq1d 4408	. . . . . . . . . 10 |- ( ( ( ( ph /\ z = Z ) /\ g = G ) /\ h = K ) -> <. z , X >. = <. Z , X >. )
24	23	oveq1d 6665	. . . . . . . . 9 |- ( ( ( ( ph /\ z = Z ) /\ g = G ) /\ h = K ) -> ( <. z , X >. .x. Y ) = ( <. Z , X >. .x. Y ) )
25		eqidd 2623	. . . . . . . . 9 |- ( ( ( ( ph /\ z = Z ) /\ g = G ) /\ h = K ) -> F = F )
26		simplr 792	. . . . . . . . 9 |- ( ( ( ( ph /\ z = Z ) /\ g = G ) /\ h = K ) -> g = G )
27	24, 25, 26	oveq123d 6671	. . . . . . . 8 |- ( ( ( ( ph /\ z = Z ) /\ g = G ) /\ h = K ) -> ( F ( <. z , X >. .x. Y ) g ) = ( F ( <. Z , X >. .x. Y ) G ) )
28		simpr 477	. . . . . . . . 9 |- ( ( ( ( ph /\ z = Z ) /\ g = G ) /\ h = K ) -> h = K )
29	24, 25, 28	oveq123d 6671	. . . . . . . 8 |- ( ( ( ( ph /\ z = Z ) /\ g = G ) /\ h = K ) -> ( F ( <. z , X >. .x. Y ) h ) = ( F ( <. Z , X >. .x. Y ) K ) )
30	27, 29	eqeq12d 2637	. . . . . . 7 |- ( ( ( ( ph /\ z = Z ) /\ g = G ) /\ h = K ) -> ( ( F ( <. z , X >. .x. Y ) g ) = ( F ( <. z , X >. .x. Y ) h ) <-> ( F ( <. Z , X >. .x. Y ) G ) = ( F ( <. Z , X >. .x. Y ) K ) ) )
31	26, 28	eqeq12d 2637	. . . . . . 7 |- ( ( ( ( ph /\ z = Z ) /\ g = G ) /\ h = K ) -> ( g = h <-> G = K ) )
32	30, 31	imbi12d 334	. . . . . 6 |- ( ( ( ( ph /\ z = Z ) /\ g = G ) /\ h = K ) -> ( ( ( F ( <. z , X >. .x. Y ) g ) = ( F ( <. z , X >. .x. Y ) h ) -> g = h ) <-> ( ( F ( <. Z , X >. .x. Y ) G ) = ( F ( <. Z , X >. .x. Y ) K ) -> G = K ) ) )
33	21, 32	rspcdv 3312	. . . . 5 |- ( ( ( ph /\ z = Z ) /\ g = G ) -> ( A. h e. ( z H X ) ( ( F ( <. z , X >. .x. Y ) g ) = ( F ( <. z , X >. .x. Y ) h ) -> g = h ) -> ( ( F ( <. Z , X >. .x. Y ) G ) = ( F ( <. Z , X >. .x. Y ) K ) -> G = K ) ) )
34	17, 33	rspcimdv 3310	. . . 4 |- ( ( ph /\ z = Z ) -> ( A. g e. ( z H X ) A. h e. ( z H X ) ( ( F ( <. z , X >. .x. Y ) g ) = ( F ( <. z , X >. .x. Y ) h ) -> g = h ) -> ( ( F ( <. Z , X >. .x. Y ) G ) = ( F ( <. Z , X >. .x. Y ) K ) -> G = K ) ) )
35	12, 34	rspcimdv 3310	. . 3 |- ( ph -> ( A. z e. B A. g e. ( z H X ) A. h e. ( z H X ) ( ( F ( <. z , X >. .x. Y ) g ) = ( F ( <. z , X >. .x. Y ) h ) -> g = h ) -> ( ( F ( <. Z , X >. .x. Y ) G ) = ( F ( <. Z , X >. .x. Y ) K ) -> G = K ) ) )
36	11, 35	mpd 15	. 2 |- ( ph -> ( ( F ( <. Z , X >. .x. Y ) G ) = ( F ( <. Z , X >. .x. Y ) K ) -> G = K ) )
37		oveq2 6658	. 2 |- ( G = K -> ( F ( <. Z , X >. .x. Y ) G ) = ( F ( <. Z , X >. .x. Y ) K ) )
38	36, 37	impbid1 215	1 |- ( ph -> ( ( F ( <. Z , X >. .x. Y ) G ) = ( F ( <. Z , X >. .x. Y ) K ) <-> G = K ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325  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-cat 16329  df-mon 16390

This theorem is referenced by:  epii  16403  monsect  16443  fthmon  16587  setcmon  16737