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Theorem issubc2 13729
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 5359 . . . . 5  |-  ( J  Fn  ( S  X.  S )  ->  dom  J  =  ( S  X.  S ) )
75, 6syl 15 . . . 4  |-  ( ph  ->  dom  J  =  ( S  X.  S ) )
87dmeqd 4897 . . 3  |-  ( ph  ->  dom  dom  J  =  dom  ( S  X.  S
) )
9 dmxpid 4914 . . 3  |-  dom  ( S  X.  S )  =  S
108, 9syl6req 2345 . 2  |-  ( ph  ->  S  =  dom  dom  J )
111, 2, 3, 4, 10issubc 13728 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656   class class class wbr 4039    X. cxp 4703   dom cdm 4705    Fn wfn 5266   ` cfv 5271  (class class class)co 5874  compcco 13236   Catccat 13582   Idccid 13583    Homf chomf 13584    C_cat cssc 13700  Subcatcsubc 13702
This theorem is referenced by:  subcidcl  13734  subccocl  13735  issubc3  13739  fullsubc  13740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pm 6791  df-ixp 6834  df-ssc 13703  df-subc 13705
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