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Theorem issubc2 15447
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 5660 . . . . 5  |-  ( J  Fn  ( S  X.  S )  ->  dom  J  =  ( S  X.  S ) )
75, 6syl 17 . . . 4  |-  ( ph  ->  dom  J  =  ( S  X.  S ) )
87dmeqd 5025 . . 3  |-  ( ph  ->  dom  dom  J  =  dom  ( S  X.  S
) )
9 dmxpid 5042 . . 3  |-  dom  ( S  X.  S )  =  S
108, 9syl6req 2460 . 2  |-  ( ph  ->  S  =  dom  dom  J )
111, 2, 3, 4, 10issubc 15446 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 367    = wceq 1405    e. wcel 1842   A.wral 2753   <.cop 3977   class class class wbr 4394    X. cxp 4820   dom cdm 4822    Fn wfn 5563   ` cfv 5568  (class class class)co 6277  compcco 14919   Catccat 15276   Idccid 15277   Hom f chomf 15278    C_cat cssc 15418  Subcatcsubc 15420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-pm 7459  df-ixp 7507  df-ssc 15421  df-subc 15423
This theorem is referenced by:  0subcat  15449  catsubcat  15450  subcidcl  15455  subccocl  15456  issubc3  15460  fullsubc  15461  rnghmsubcsetc  38277  rhmsubcsetc  38323  rhmsubcrngc  38329  srhmsubc  38376  rhmsubc  38390  srhmsubcALTV  38395  rhmsubcALTV  38409
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