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Theorem issubc2 15055
Description: Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
issubc.h  |-  H  =  ( Hom f  `  C )
issubc.i  |-  .1.  =  ( Id `  C )
issubc.o  |-  .x.  =  (comp `  C )
issubc.c  |-  ( ph  ->  C  e.  Cat )
issubc2.a  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
Assertion
Ref Expression
issubc2  |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  H  /\  A. x  e.  S  ( (  .1.  `  x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x J z ) ) ) ) )
Distinct variable groups:    f, g, x, y, z, C    f, J, g, x, y, z    S, f, g, x, y, z
Allowed substitution hints:    ph( x, y, z, f, g)    .x. ( x, y, z, f, g)    .1. ( x, y, z, f, g)    H( x, y, z, f, g)

Proof of Theorem issubc2
StepHypRef Expression
1 issubc.h . 2  |-  H  =  ( Hom f  `  C )
2 issubc.i . 2  |-  .1.  =  ( Id `  C )
3 issubc.o . 2  |-  .x.  =  (comp `  C )
4 issubc.c . 2  |-  ( ph  ->  C  e.  Cat )
5 issubc2.a . . . . 5  |-  ( ph  ->  J  Fn  ( S  X.  S ) )
6 fndm 5671 . . . . 5  |-  ( J  Fn  ( S  X.  S )  ->  dom  J  =  ( S  X.  S ) )
75, 6syl 16 . . . 4  |-  ( ph  ->  dom  J  =  ( S  X.  S ) )
87dmeqd 5196 . . 3  |-  ( ph  ->  dom  dom  J  =  dom  ( S  X.  S
) )
9 dmxpid 5213 . . 3  |-  dom  ( S  X.  S )  =  S
108, 9syl6req 2518 . 2  |-  ( ph  ->  S  =  dom  dom  J )
111, 2, 3, 4, 10issubc 15054 1  |-  ( ph  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  H  /\  A. x  e.  S  ( (  .1.  `  x )  e.  ( x J x )  /\  A. y  e.  S  A. z  e.  S  A. f  e.  ( x J y ) A. g  e.  ( y J z ) ( g (
<. x ,  y >.  .x.  z ) f )  e.  ( x J z ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   <.cop 4026   class class class wbr 4440    X. cxp 4990   dom cdm 4992    Fn wfn 5574   ` cfv 5579  (class class class)co 6275  compcco 14556   Catccat 14908   Idccid 14909   Hom f chomf 14910    C_cat cssc 15026  Subcatcsubc 15028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-pm 7413  df-ixp 7460  df-ssc 15029  df-subc 15031
This theorem is referenced by:  subcidcl  15060  subccocl  15061  issubc3  15065  fullsubc  15066
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