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Theorem sectmon 13680
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 2283 . . . . 5  |-  (  Hom  `  C )  =  (  Hom  `  C )
4 eqid 2283 . . . . 5  |-  (comp `  C )  =  (comp `  C )
5 eqid 2283 . . . . 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 13656 . . . 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 201 . . 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 967 . 2  |-  ( ph  ->  F  e.  ( X (  Hom  `  C
) Y ) )
13 oveq2 5866 . . . . 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 969 . . . . . . . . 9  |-  ( ph  ->  ( G ( <. X ,  Y >. (comp `  C ) X ) F )  =  ( ( Id `  C
) `  X )
)
1514ad2antrr 706 . . . . . . . 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 5873 . . . . . . 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 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  C  e.  Cat )
18 simplr 731 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  x  e.  B
)
198ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  X  e.  B
)
209ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  Y  e.  B
)
21 simprl 732 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  g  e.  ( x (  Hom  `  C
) X ) )
2212ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  (
g  e.  ( x (  Hom  `  C
) X )  /\  h  e.  ( x
(  Hom  `  C ) X ) ) )  ->  F  e.  ( X (  Hom  `  C
) Y ) )
2311simp2d 968 . . . . . . . . 9  |-  ( ph  ->  G  e.  ( Y (  Hom  `  C
) X ) )
2423ad2antrr 706 . . . . . . . 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 13588 . . . . . . 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 13585 . . . . . . 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 2323 . . . . . 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 5873 . . . . . . 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 733 . . . . . . . 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 13588 . . . . . . 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 13585 . . . . . . 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 2323 . . . . . 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 2297 . . . . 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 210 . . . 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 2635 . . 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 2626 . 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 13637 . 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 888 1  |-  ( ph  ->  F  e.  ( X M Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567  Monocmon 13631  Sectcsect 13647
This theorem is referenced by:  sectepi  13682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-cat 13570  df-cid 13571  df-mon 13633  df-sect 13650
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