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Theorem issubc2 14038
Description: Elementhood in the set of subcategories. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
issubc.h  |-  H  =  (  Homf 
`  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  =  (  Homf 
`  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 5546 . . . . 5  |-  ( J  Fn  ( S  X.  S )  ->  dom  J  =  ( S  X.  S ) )
75, 6syl 16 . . . 4  |-  ( ph  ->  dom  J  =  ( S  X.  S ) )
87dmeqd 5074 . . 3  |-  ( ph  ->  dom  dom  J  =  dom  ( S  X.  S
) )
9 dmxpid 5091 . . 3  |-  dom  ( S  X.  S )  =  S
108, 9syl6req 2487 . 2  |-  ( ph  ->  S  =  dom  dom  J )
111, 2, 3, 4, 10issubc 14037 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   <.cop 3819   class class class wbr 4214    X. cxp 4878   dom cdm 4880    Fn wfn 5451   ` cfv 5456  (class class class)co 6083  compcco 13543   Catccat 13891   Idccid 13892    Homf chomf 13893    C_cat cssc 14009  Subcatcsubc 14011
This theorem is referenced by:  subcidcl  14043  subccocl  14044  issubc3  14048  fullsubc  14049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-pm 7023  df-ixp 7066  df-ssc 14012  df-subc 14014
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