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Theorem subsubc 14762
Description: A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypothesis
Ref Expression
subsubc.d  |-  D  =  ( C  |`cat  H )
Assertion
Ref Expression
subsubc  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H ) ) )

Proof of Theorem subsubc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . 6  |-  ( J  e.  (Subcat `  D
)  ->  J  e.  (Subcat `  D ) )
2 eqid 2442 . . . . . 6  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
31, 2subcssc 14749 . . . . 5  |-  ( J  e.  (Subcat `  D
)  ->  J  C_cat  ( Hom f  `  D ) )
4 subsubc.d . . . . . . 7  |-  D  =  ( C  |`cat  H )
5 eqid 2442 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
6 subcrcl 14728 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  Cat )
7 id 22 . . . . . . . 8  |-  ( H  e.  (Subcat `  C
)  ->  H  e.  (Subcat `  C ) )
8 eqidd 2443 . . . . . . . 8  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  =  dom  dom  H )
97, 8subcfn 14750 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  H  Fn  ( dom  dom  H  X.  dom  dom  H ) )
107, 9, 5subcss1 14751 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  C_  ( Base `  C
) )
114, 5, 6, 9, 10reschomf 14743 . . . . . 6  |-  ( H  e.  (Subcat `  C
)  ->  H  =  ( Hom f  `  D ) )
1211breq2d 4303 . . . . 5  |-  ( H  e.  (Subcat `  C
)  ->  ( J  C_cat  H  <-> 
J  C_cat  ( Hom f  `  D ) ) )
133, 12syl5ibr 221 . . . 4  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  ->  J  C_cat  H ) )
1413pm4.71rd 635 . . 3  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  D )
) ) )
15 simpr 461 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  H )
16 simpl 457 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  e.  (Subcat `  C ) )
17 eqid 2442 . . . . . . . . 9  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
1816, 17subcssc 14749 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  C_cat  ( Hom f  `  C ) )
19 ssctr 14737 . . . . . . . 8  |-  ( ( J  C_cat  H  /\  H  C_cat  ( Hom f  `  C ) )  ->  J  C_cat  ( Hom f  `  C ) )
2015, 18, 19syl2anc 661 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  ( Hom f  `  C ) )
2112biimpa 484 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  ( Hom f  `  D ) )
2220, 212thd 240 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  C_cat  ( Hom f  `  C )  <->  J  C_cat  ( Hom f  `  D ) ) )
2316adantr 465 . . . . . . . . 9  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  ->  H  e.  (Subcat `  C
) )
249adantr 465 . . . . . . . . . 10  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  Fn  ( dom  dom  H  X.  dom  dom  H ) )
2524adantr 465 . . . . . . . . 9  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  ->  H  Fn  ( dom  dom 
H  X.  dom  dom  H ) )
26 eqid 2442 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
27 eqidd 2443 . . . . . . . . . . . 12  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  J  =  dom  dom  J )
2815, 27sscfn1 14729 . . . . . . . . . . 11  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  Fn  ( dom  dom  J  X.  dom  dom  J ) )
2928, 24, 15ssc1 14733 . . . . . . . . . 10  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  J  C_ 
dom  dom  H )
3029sselda 3355 . . . . . . . . 9  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  ->  x  e.  dom  dom  H
)
314, 23, 25, 26, 30subcid 14756 . . . . . . . 8  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  -> 
( ( Id `  C ) `  x
)  =  ( ( Id `  D ) `
 x ) )
3231eleq1d 2508 . . . . . . 7  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  -> 
( ( ( Id
`  C ) `  x )  e.  ( x J x )  <-> 
( ( Id `  D ) `  x
)  e.  ( x J x ) ) )
3332ralbidva 2730 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( A. x  e.  dom  dom  J
( ( Id `  C ) `  x
)  e.  ( x J x )  <->  A. x  e.  dom  dom  J (
( Id `  D
) `  x )  e.  ( x J x ) ) )
344oveq1i 6100 . . . . . . . 8  |-  ( D  |`cat 
J )  =  ( ( C  |`cat  H )  |`cat  J )
356adantr 465 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  C  e.  Cat )
36 dmexg 6508 . . . . . . . . . . 11  |-  ( H  e.  (Subcat `  C
)  ->  dom  H  e. 
_V )
37 dmexg 6508 . . . . . . . . . . 11  |-  ( dom 
H  e.  _V  ->  dom 
dom  H  e.  _V )
3836, 37syl 16 . . . . . . . . . 10  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  e.  _V )
3938adantr 465 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  H  e.  _V )
4035, 24, 28, 39, 29rescabs 14745 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( ( C  |`cat  H )  |`cat  J )  =  ( C  |`cat  J
) )
4134, 40syl5req 2487 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( C  |`cat  J )  =  ( D  |`cat 
J ) )
4241eleq1d 2508 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( ( C  |`cat  J )  e.  Cat  <->  ( D  |`cat  J )  e.  Cat ) )
4322, 33, 423anbi123d 1289 . . . . 5  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( ( J  C_cat  ( Hom f  `  C )  /\  A. x  e.  dom  dom  J ( ( Id `  C ) `  x
)  e.  ( x J x )  /\  ( C  |`cat  J )  e.  Cat ) 
<->  ( J  C_cat  ( Hom f  `  D )  /\  A. x  e.  dom  dom  J
( ( Id `  D ) `  x
)  e.  ( x J x )  /\  ( D  |`cat  J )  e.  Cat ) ) )
44 eqid 2442 . . . . . 6  |-  ( C  |`cat 
J )  =  ( C  |`cat  J )
4517, 26, 44, 35, 28issubc3 14758 . . . . 5  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  ( Hom f  `  C )  /\  A. x  e.  dom  dom  J ( ( Id `  C ) `  x
)  e.  ( x J x )  /\  ( C  |`cat  J )  e.  Cat ) ) )
46 eqid 2442 . . . . . 6  |-  ( Id
`  D )  =  ( Id `  D
)
47 eqid 2442 . . . . . 6  |-  ( D  |`cat 
J )  =  ( D  |`cat  J )
484, 7subccat 14757 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  D  e.  Cat )
4948adantr 465 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  D  e.  Cat )
502, 46, 47, 49, 28issubc3 14758 . . . . 5  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  e.  (Subcat `  D )  <->  ( J  C_cat  ( Hom f  `  D )  /\  A. x  e.  dom  dom  J ( ( Id `  D ) `  x
)  e.  ( x J x )  /\  ( D  |`cat  J )  e.  Cat ) ) )
5143, 45, 503bitr4rd 286 . . . 4  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  e.  (Subcat `  D )  <->  J  e.  (Subcat `  C
) ) )
5251pm5.32da 641 . . 3  |-  ( H  e.  (Subcat `  C
)  ->  ( ( J  C_cat  H  /\  J  e.  (Subcat `  D )
)  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  C ) ) ) )
5314, 52bitrd 253 . 2  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  C )
) ) )
54 ancom 450 . 2  |-  ( ( J  C_cat  H  /\  J  e.  (Subcat `  C )
)  <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H )
)
5553, 54syl6bb 261 1  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2714   _Vcvv 2971   class class class wbr 4291    X. cxp 4837   dom cdm 4839    Fn wfn 5412   ` cfv 5417  (class class class)co 6090   Basecbs 14173   Catccat 14601   Idccid 14602   Hom f chomf 14603    C_cat cssc 14719    |`cat cresc 14720  Subcatcsubc 14721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-er 7100  df-pm 7216  df-ixp 7263  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-2 10379  df-3 10380  df-4 10381  df-5 10382  df-6 10383  df-7 10384  df-8 10385  df-9 10386  df-10 10387  df-n0 10579  df-z 10646  df-dec 10755  df-ndx 14176  df-slot 14177  df-base 14178  df-sets 14179  df-ress 14180  df-hom 14261  df-cco 14262  df-cat 14605  df-cid 14606  df-homf 14607  df-ssc 14722  df-resc 14723  df-subc 14724
This theorem is referenced by: (None)
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