Theorem ismon 16393

Description: Definition of a monomorphism in a category. (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 )

Assertion

Ref	Expression
ismon	|- ( ph -> ( F e. ( X M Y ) <-> ( F e. ( X H Y ) /\ A. z e. B Fun `' ( g e. ( z H X ) |-> ( F ( <. z , X >. .x. Y ) g ) ) ) ) )

Distinct variable groups:   z, g, B   ph, g, z   C, g, z   g, H, z   .x. , g, z   g, F, z   g, X, z   g, Y, z

Allowed substitution hints:   M( z, g)

Proof of Theorem ismon

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

Step	Hyp	Ref	Expression
1		ismon.b	. . . . 5 |- B = ( Base ` C )
2		ismon.h	. . . . 5 |- H = ( Hom ` C )
3		ismon.o	. . . . 5 |- .x. = (comp ` C )
4		ismon.s	. . . . 5 |- M = (Mono ` C )
5		ismon.c	. . . . 5 |- ( ph -> C e. Cat )
6	1, 2, 3, 4, 5	monfval 16392	. . . 4 |- ( ph -> M = ( x e. B , y e. B |-> { f e. ( x H y ) | A. z e. B Fun `' ( g e. ( z H x ) |-> ( f ( <. z , x >. .x. y ) g ) ) } ) )
7		simprl 794	. . . . . 6 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> x = X )
8		simprr 796	. . . . . 6 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> y = Y )
9	7, 8	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x H y ) = ( X H Y ) )
10	7	oveq2d 6666	. . . . . . . . 9 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( z H x ) = ( z H X ) )
11	7	opeq2d 4409	. . . . . . . . . . 11 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> <. z , x >. = <. z , X >. )
12	11, 8	oveq12d 6668	. . . . . . . . . 10 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( <. z , x >. .x. y ) = ( <. z , X >. .x. Y ) )
13	12	oveqd 6667	. . . . . . . . 9 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( f ( <. z , x >. .x. y ) g ) = ( f ( <. z , X >. .x. Y ) g ) )
14	10, 13	mpteq12dv 4733	. . . . . . . 8 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( g e. ( z H x ) |-> ( f ( <. z , x >. .x. y ) g ) ) = ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) )
15	14	cnveqd 5298	. . . . . . 7 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> `' ( g e. ( z H x ) |-> ( f ( <. z , x >. .x. y ) g ) ) = `' ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) )
16	15	funeqd 5910	. . . . . 6 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( Fun `' ( g e. ( z H x ) |-> ( f ( <. z , x >. .x. y ) g ) ) <-> Fun `' ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) ) )
17	16	ralbidv 2986	. . . . 5 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( A. z e. B Fun `' ( g e. ( z H x ) |-> ( f ( <. z , x >. .x. y ) g ) ) <-> A. z e. B Fun `' ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) ) )
18	9, 17	rabeqbidv 3195	. . . 4 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> { f e. ( x H y ) | A. z e. B Fun `' ( g e. ( z H x ) |-> ( f ( <. z , x >. .x. y ) g ) ) } = { f e. ( X H Y ) | A. z e. B Fun `' ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) } )
19		ismon.x	. . . 4 |- ( ph -> X e. B )
20		ismon.y	. . . 4 |- ( ph -> Y e. B )
21		ovex 6678	. . . . . 6 |- ( X H Y ) e. _V
22	21	rabex 4813	. . . . 5 |- { f e. ( X H Y ) | A. z e. B Fun `' ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) } e. _V
23	22	a1i 11	. . . 4 |- ( ph -> { f e. ( X H Y ) | A. z e. B Fun `' ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) } e. _V )
24	6, 18, 19, 20, 23	ovmpt2d 6788	. . 3 |- ( ph -> ( X M Y ) = { f e. ( X H Y ) | A. z e. B Fun `' ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) } )
25	24	eleq2d 2687	. 2 |- ( ph -> ( F e. ( X M Y ) <-> F e. { f e. ( X H Y ) | A. z e. B Fun `' ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) } ) )
26		oveq1 6657	. . . . . . 7 |- ( f = F -> ( f ( <. z , X >. .x. Y ) g ) = ( F ( <. z , X >. .x. Y ) g ) )
27	26	mpteq2dv 4745	. . . . . 6 |- ( f = F -> ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) = ( g e. ( z H X ) |-> ( F ( <. z , X >. .x. Y ) g ) ) )
28	27	cnveqd 5298	. . . . 5 |- ( f = F -> `' ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) = `' ( g e. ( z H X ) |-> ( F ( <. z , X >. .x. Y ) g ) ) )
29	28	funeqd 5910	. . . 4 |- ( f = F -> ( Fun `' ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) <-> Fun `' ( g e. ( z H X ) |-> ( F ( <. z , X >. .x. Y ) g ) ) ) )
30	29	ralbidv 2986	. . 3 |- ( f = F -> ( A. z e. B Fun `' ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) <-> A. z e. B Fun `' ( g e. ( z H X ) |-> ( F ( <. z , X >. .x. Y ) g ) ) ) )
31	30	elrab 3363	. 2 |- ( F e. { f e. ( X H Y ) | A. z e. B Fun `' ( g e. ( z H X ) |-> ( f ( <. z , X >. .x. Y ) g ) ) } <-> ( F e. ( X H Y ) /\ A. z e. B Fun `' ( g e. ( z H X ) |-> ( F ( <. z , X >. .x. Y ) g ) ) ) )
32	25, 31	syl6bb 276	1 |- ( ph -> ( F e. ( X M Y ) <-> ( F e. ( X H Y ) /\ A. z e. B Fun `' ( g e. ( z H X ) |-> ( F ( <. z , X >. .x. Y ) g ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   <.cop 4183   |-> cmpt 4729   `'ccnv 5113   Fun wfun 5882   ` 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-mon 16390

This theorem is referenced by:  ismon2  16394  monhom  16395  isepi  16400