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Theorem sectmon 13958
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.
StepHypRef Expression
1 sectmon.1 . . . 4  |-  ( ph  ->  F ( X S Y ) G )
2 sectmon.b . . . . 5  |-  B  =  ( Base `  C
)
3 eqid 2404 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2404 . . . . 5  |-  (comp `  C )  =  (comp `  C )
5 eqid 2404 . . . . 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 )
102, 3, 4, 5, 6, 7, 8, 9issect 13934 . . . 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 ) ) ) )
111, 10mpbid 202 . . 3  |-  ( ph  ->  ( F  e.  ( X (  Hom  `  C
) Y )  /\  G  e.  ( Y
(  Hom  `  C ) X )  /\  ( G ( <. X ,  Y >. (comp `  C
) X ) F )  =  ( ( Id `  C ) `
 X ) ) )
1211simp1d 969 . 2  |-  ( ph  ->  F  e.  ( X (  Hom  `  C
) Y ) )
13 oveq2 6048 . . . . 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 ) ) )
1411simp3d 971 . . . . . . . . 9  |-  ( ph  ->  ( G ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
)
1514ad2antrr 707 . . . . . . . 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 )
)
1615oveq1d 6055 . . . . . . 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 ) )
177ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  C  e.  Cat )
18 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  x  e.  B
)
198ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  X  e.  B
)
209ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  Y  e.  B
)
21 simprl 733 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  g  e.  ( x (  Hom  `  C
) X ) )
2212ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  F  e.  ( X (  Hom  `  C
) Y ) )
2311simp2d 970 . . . . . . . . 9  |-  ( ph  ->  G  e.  ( Y (  Hom  `  C
) X ) )
2423ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  G  e.  ( Y (  Hom  `  C
) X ) )
252, 3, 4, 17, 18, 19, 20, 21, 22, 19, 24catass 13866 . . . . . . 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 ) ) )
262, 3, 5, 17, 18, 4, 19, 21catlid 13863 . . . . . . 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 )
2716, 25, 263eqtr3d 2444 . . . . . 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 )
2815oveq1d 6055 . . . . . . 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 734 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  h  e.  ( x (  Hom  `  C
) X ) )
302, 3, 4, 17, 18, 19, 20, 29, 22, 19, 24catass 13866 . . . . . . 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 ) ) )
312, 3, 5, 17, 18, 4, 19, 29catlid 13863 . . . . . . 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 )
3228, 30, 313eqtr3d 2444 . . . . . 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 )
3327, 32eqeq12d 2418 . . . . 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 ) )
3413, 33syl5ib 211 . . . 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 ) )
3534ralrimivva 2758 . . 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 ) )
3635ralrimiva 2749 . 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 )
382, 3, 4, 37, 7, 8, 9ismon2 13915 . 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 ) ) ) )
3912, 36, 38mpbir2and 889 1  |-  ( ph  ->  F  e.  ( X M Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   <.cop 3777   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844   Idccid 13845  Monocmon 13909  Sectcsect 13925
This theorem is referenced by:  sectepi  13960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-cat 13848  df-cid 13849  df-mon 13911  df-sect 13928
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