Theorem sectmon 16442

Description: If F is a section of G, then F is a monomorphism. A monomorphism that arises from a section is also known as a split monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
sectmon.b	|- B = ( Base ` C )
sectmon.m	|- M = (Mono ` C )
sectmon.s	|- S = (Sect ` C )
sectmon.c	|- ( ph -> C e. Cat )
sectmon.x	|- ( ph -> X e. B )
sectmon.y	|- ( ph -> Y e. B )
sectmon.1	|- ( ph -> F ( X S Y ) G )

Assertion

Ref	Expression
sectmon	|- ( ph -> F e. ( X M Y ) )

Proof of Theorem sectmon

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

Step	Hyp	Ref	Expression
1		sectmon.1	. . . 4 |- ( ph -> F ( X S Y ) G )
2		sectmon.b	. . . . 5 |- B = ( Base ` C )
3		eqid 2622	. . . . 5 |- ( Hom ` C ) = ( Hom ` C )
4		eqid 2622	. . . . 5 |- (comp ` C ) = (comp ` C )
5		eqid 2622	. . . . 5 |- ( Id ` C ) = ( Id ` C )
6		sectmon.s	. . . . 5 |- S = (Sect ` C )
7		sectmon.c	. . . . 5 |- ( ph -> C e. Cat )
8		sectmon.x	. . . . 5 |- ( ph -> X e. B )
9		sectmon.y	. . . . 5 |- ( ph -> Y e. B )
10	2, 3, 4, 5, 6, 7, 8, 9	issect 16413	. . . 4 |- ( ph -> ( F ( X S Y ) G <-> ( F e. ( X ( Hom ` C ) Y ) /\ G e. ( Y ( Hom ` C ) X ) /\ ( G ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) ) )
11	1, 10	mpbid 222	. . 3 |- ( ph -> ( F e. ( X ( Hom ` C ) Y ) /\ G e. ( Y ( Hom ` C ) X ) /\ ( G ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) ) )
12	11	simp1d 1073	. 2 |- ( ph -> F e. ( X ( Hom ` C ) Y ) )
13		oveq2 6658	. . . . 5 |- ( ( F ( <. x , X >. (comp ` C ) Y ) g ) = ( F ( <. x , X >. (comp ` C ) Y ) h ) -> ( G ( <. x , Y >. (comp ` C ) X ) ( F ( <. x , X >. (comp ` C ) Y ) g ) ) = ( G ( <. x , Y >. (comp ` C ) X ) ( F ( <. x , X >. (comp ` C ) Y ) h ) ) )
14	11	simp3d 1075	. . . . . . . . 9 |- ( ph -> ( G ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) )
15	14	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> ( G ( <. X , Y >. (comp ` C ) X ) F ) = ( ( Id ` C ) ` X ) )
16	15	oveq1d 6665	. . . . . . 7 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> ( ( G ( <. X , Y >. (comp ` C ) X ) F ) ( <. x , X >. (comp ` C ) X ) g ) = ( ( ( Id ` C ) ` X ) ( <. x , X >. (comp ` C ) X ) g ) )
17	7	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> C e. Cat )
18		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> x e. B )
19	8	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> X e. B )
20	9	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> Y e. B )
21		simprl 794	. . . . . . . 8 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> g e. ( x ( Hom ` C ) X ) )
22	12	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> F e. ( X ( Hom ` C ) Y ) )
23	11	simp2d 1074	. . . . . . . . 9 |- ( ph -> G e. ( Y ( Hom ` C ) X ) )
24	23	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> G e. ( Y ( Hom ` C ) X ) )
25	2, 3, 4, 17, 18, 19, 20, 21, 22, 19, 24	catass 16347	. . . . . . 7 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> ( ( G ( <. X , Y >. (comp ` C ) X ) F ) ( <. x , X >. (comp ` C ) X ) g ) = ( G ( <. x , Y >. (comp ` C ) X ) ( F ( <. x , X >. (comp ` C ) Y ) g ) ) )
26	2, 3, 5, 17, 18, 4, 19, 21	catlid 16344	. . . . . . 7 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> ( ( ( Id ` C ) ` X ) ( <. x , X >. (comp ` C ) X ) g ) = g )
27	16, 25, 26	3eqtr3d 2664	. . . . . 6 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> ( G ( <. x , Y >. (comp ` C ) X ) ( F ( <. x , X >. (comp ` C ) Y ) g ) ) = g )
28	15	oveq1d 6665	. . . . . . 7 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> ( ( G ( <. X , Y >. (comp ` C ) X ) F ) ( <. x , X >. (comp ` C ) X ) h ) = ( ( ( Id ` C ) ` X ) ( <. x , X >. (comp ` C ) X ) h ) )
29		simprr 796	. . . . . . . 8 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> h e. ( x ( Hom ` C ) X ) )
30	2, 3, 4, 17, 18, 19, 20, 29, 22, 19, 24	catass 16347	. . . . . . 7 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> ( ( G ( <. X , Y >. (comp ` C ) X ) F ) ( <. x , X >. (comp ` C ) X ) h ) = ( G ( <. x , Y >. (comp ` C ) X ) ( F ( <. x , X >. (comp ` C ) Y ) h ) ) )
31	2, 3, 5, 17, 18, 4, 19, 29	catlid 16344	. . . . . . 7 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> ( ( ( Id ` C ) ` X ) ( <. x , X >. (comp ` C ) X ) h ) = h )
32	28, 30, 31	3eqtr3d 2664	. . . . . 6 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> ( G ( <. x , Y >. (comp ` C ) X ) ( F ( <. x , X >. (comp ` C ) Y ) h ) ) = h )
33	27, 32	eqeq12d 2637	. . . . 5 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> ( ( G ( <. x , Y >. (comp ` C ) X ) ( F ( <. x , X >. (comp ` C ) Y ) g ) ) = ( G ( <. x , Y >. (comp ` C ) X ) ( F ( <. x , X >. (comp ` C ) Y ) h ) ) <-> g = h ) )
34	13, 33	syl5ib 234	. . . 4 |- ( ( ( ph /\ x e. B ) /\ ( g e. ( x ( Hom ` C ) X ) /\ h e. ( x ( Hom ` C ) X ) ) ) -> ( ( F ( <. x , X >. (comp ` C ) Y ) g ) = ( F ( <. x , X >. (comp ` C ) Y ) h ) -> g = h ) )
35	34	ralrimivva 2971	. . 3 |- ( ( ph /\ x e. B ) -> A. g e. ( x ( Hom ` C ) X ) A. h e. ( x ( Hom ` C ) X ) ( ( F ( <. x , X >. (comp ` C ) Y ) g ) = ( F ( <. x , X >. (comp ` C ) Y ) h ) -> g = h ) )
36	35	ralrimiva 2966	. 2 |- ( ph -> A. x e. B A. g e. ( x ( Hom ` C ) X ) A. h e. ( x ( Hom ` C ) X ) ( ( F ( <. x , X >. (comp ` C ) Y ) g ) = ( F ( <. x , X >. (comp ` C ) Y ) h ) -> g = h ) )
37		sectmon.m	. . 3 |- M = (Mono ` C )
38	2, 3, 4, 37, 7, 8, 9	ismon2 16394	. 2 |- ( ph -> ( F e. ( X M Y ) <-> ( F e. ( X ( Hom ` C ) Y ) /\ A. x e. B A. g e. ( x ( Hom ` C ) X ) A. h e. ( x ( Hom ` C ) X ) ( ( F ( <. x , X >. (comp ` C ) Y ) g ) = ( F ( <. x , X >. (comp ` C ) Y ) h ) -> g = h ) ) ) )
39	12, 36, 38	mpbir2and 957	1 |- ( ph -> F 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   Idccid 16326  Monocmon 16388  Sectcsect 16404

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-rmo 2920  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cat 16329  df-cid 16330  df-mon 16390  df-sect 16407

This theorem is referenced by:  sectepi  16444