Theorem monfval 16392

Description: Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-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 )

Assertion

Ref	Expression
monfval	|- ( 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 ) ) } ) )

Distinct variable groups:   f, g, x, y, z, B   ph, f, g, x, y, z   C, f, g, x, y, z   f, H, g, x, y, z   .x. , f, g, x, y, z   f, M

Allowed substitution hints:   M( x, y, z, g)

Proof of Theorem monfval

Dummy variables b c h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismon.s	. 2 |- M = (Mono ` C )
2		ismon.c	. . 3 |- ( ph -> C e. Cat )
3		fvexd 6203	. . . . 5 |- ( c = C -> ( Base ` c ) e. _V )
4		fveq2 6191	. . . . . 6 |- ( c = C -> ( Base ` c ) = ( Base ` C ) )
5		ismon.b	. . . . . 6 |- B = ( Base ` C )
6	4, 5	syl6eqr 2674	. . . . 5 |- ( c = C -> ( Base ` c ) = B )
7		fvexd 6203	. . . . . 6 |- ( ( c = C /\ b = B ) -> ( Hom ` c ) e. _V )
8		simpl 473	. . . . . . . 8 |- ( ( c = C /\ b = B ) -> c = C )
9	8	fveq2d 6195	. . . . . . 7 |- ( ( c = C /\ b = B ) -> ( Hom ` c ) = ( Hom ` C ) )
10		ismon.h	. . . . . . 7 |- H = ( Hom ` C )
11	9, 10	syl6eqr 2674	. . . . . 6 |- ( ( c = C /\ b = B ) -> ( Hom ` c ) = H )
12		simplr 792	. . . . . . 7 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> b = B )
13		simpr 477	. . . . . . . . 9 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> h = H )
14	13	oveqd 6667	. . . . . . . 8 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> ( x h y ) = ( x H y ) )
15	13	oveqd 6667	. . . . . . . . . . . 12 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> ( z h x ) = ( z H x ) )
16		simpll 790	. . . . . . . . . . . . . . . 16 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> c = C )
17	16	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> (comp ` c ) = (comp ` C ) )
18		ismon.o	. . . . . . . . . . . . . . 15 |- .x. = (comp ` C )
19	17, 18	syl6eqr 2674	. . . . . . . . . . . . . 14 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> (comp ` c ) = .x. )
20	19	oveqd 6667	. . . . . . . . . . . . 13 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> ( <. z , x >. (comp ` c ) y ) = ( <. z , x >. .x. y ) )
21	20	oveqd 6667	. . . . . . . . . . . 12 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> ( f ( <. z , x >. (comp ` c ) y ) g ) = ( f ( <. z , x >. .x. y ) g ) )
22	15, 21	mpteq12dv 4733	. . . . . . . . . . 11 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> ( g e. ( z h x ) |-> ( f ( <. z , x >. (comp ` c ) y ) g ) ) = ( g e. ( z H x ) |-> ( f ( <. z , x >. .x. y ) g ) ) )
23	22	cnveqd 5298	. . . . . . . . . 10 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> `' ( g e. ( z h x ) |-> ( f ( <. z , x >. (comp ` c ) y ) g ) ) = `' ( g e. ( z H x ) |-> ( f ( <. z , x >. .x. y ) g ) ) )
24	23	funeqd 5910	. . . . . . . . 9 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> ( Fun `' ( g e. ( z h x ) |-> ( f ( <. z , x >. (comp ` c ) y ) g ) ) <-> Fun `' ( g e. ( z H x ) |-> ( f ( <. z , x >. .x. y ) g ) ) ) )
25	12, 24	raleqbidv 3152	. . . . . . . 8 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> ( A. z e. b Fun `' ( g e. ( z h x ) |-> ( f ( <. z , x >. (comp ` c ) y ) g ) ) <-> A. z e. B Fun `' ( g e. ( z H x ) |-> ( f ( <. z , x >. .x. y ) g ) ) ) )
26	14, 25	rabeqbidv 3195	. . . . . . 7 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> { f e. ( x h y ) | A. z e. b Fun `' ( g e. ( z h x ) |-> ( f ( <. z , x >. (comp ` c ) y ) g ) ) } = { f e. ( x H y ) | A. z e. B Fun `' ( g e. ( z H x ) |-> ( f ( <. z , x >. .x. y ) g ) ) } )
27	12, 12, 26	mpt2eq123dv 6717	. . . . . 6 |- ( ( ( c = C /\ b = B ) /\ h = H ) -> ( x e. b , y e. b |-> { f e. ( x h y ) | A. z e. b Fun `' ( g e. ( z h x ) |-> ( f ( <. z , x >. (comp ` c ) y ) g ) ) } ) = ( 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 ) ) } ) )
28	7, 11, 27	csbied2 3561	. . . . 5 |- ( ( c = C /\ b = B ) -> [_ ( Hom ` c ) / h ]_ ( x e. b , y e. b |-> { f e. ( x h y ) | A. z e. b Fun `' ( g e. ( z h x ) |-> ( f ( <. z , x >. (comp ` c ) y ) g ) ) } ) = ( 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 ) ) } ) )
29	3, 6, 28	csbied2 3561	. . . 4 |- ( c = C -> [_ ( Base ` c ) / b ]_ [_ ( Hom ` c ) / h ]_ ( x e. b , y e. b |-> { f e. ( x h y ) | A. z e. b Fun `' ( g e. ( z h x ) |-> ( f ( <. z , x >. (comp ` c ) y ) g ) ) } ) = ( 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 ) ) } ) )
30		df-mon 16390	. . . 4 |- Mono = ( c e. Cat |-> [_ ( Base ` c ) / b ]_ [_ ( Hom ` c ) / h ]_ ( x e. b , y e. b |-> { f e. ( x h y ) | A. z e. b Fun `' ( g e. ( z h x ) |-> ( f ( <. z , x >. (comp ` c ) y ) g ) ) } ) )
31		fvex 6201	. . . . . 6 |- ( Base ` C ) e. _V
32	5, 31	eqeltri 2697	. . . . 5 |- B e. _V
33	32, 32	mpt2ex 7247	. . . 4 |- ( 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 ) ) } ) e. _V
34	29, 30, 33	fvmpt 6282	. . 3 |- ( C e. Cat -> (Mono ` C ) = ( 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 ) ) } ) )
35	2, 34	syl 17	. 2 |- ( ph -> (Mono ` C ) = ( 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 ) ) } ) )
36	1, 35	syl5eq 2668	1 |- ( 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 ) ) } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   [_csb 3533   <.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  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:  ismon  16393  monpropd  16397