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Theorem subsubc 15751
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 2450 . . . . . 6  |-  ( Hom f  `  D )  =  ( Hom f  `  D )
31, 2subcssc 15738 . . . . 5  |-  ( J  e.  (Subcat `  D
)  ->  J  C_cat  ( Hom f  `  D ) )
4 subsubc.d . . . . . . 7  |-  D  =  ( C  |`cat  H )
5 eqid 2450 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
6 subcrcl 15714 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  Cat )
7 id 22 . . . . . . . 8  |-  ( H  e.  (Subcat `  C
)  ->  H  e.  (Subcat `  C ) )
8 eqidd 2451 . . . . . . . 8  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  =  dom  dom  H )
97, 8subcfn 15739 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  H  Fn  ( dom  dom  H  X.  dom  dom  H ) )
107, 9, 5subcss1 15740 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  C_  ( Base `  C
) )
114, 5, 6, 9, 10reschomf 15729 . . . . . 6  |-  ( H  e.  (Subcat `  C
)  ->  H  =  ( Hom f  `  D ) )
1211breq2d 4413 . . . . 5  |-  ( H  e.  (Subcat `  C
)  ->  ( J  C_cat  H  <-> 
J  C_cat  ( Hom f  `  D ) ) )
133, 12syl5ibr 225 . . . 4  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  ->  J  C_cat  H ) )
1413pm4.71rd 640 . . 3  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  D )
) ) )
15 simpr 463 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  H )
16 simpl 459 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  e.  (Subcat `  C ) )
17 eqid 2450 . . . . . . . . 9  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
1816, 17subcssc 15738 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  C_cat  ( Hom f  `  C ) )
19 ssctr 15723 . . . . . . . 8  |-  ( ( J  C_cat  H  /\  H  C_cat  ( Hom f  `  C ) )  ->  J  C_cat  ( Hom f  `  C ) )
2015, 18, 19syl2anc 666 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  ( Hom f  `  C ) )
2112biimpa 487 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  ( Hom f  `  D ) )
2220, 212thd 244 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  C_cat  ( Hom f  `  C )  <->  J  C_cat  ( Hom f  `  D ) ) )
2316adantr 467 . . . . . . . . 9  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  ->  H  e.  (Subcat `  C
) )
249adantr 467 . . . . . . . . . 10  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  Fn  ( dom  dom  H  X.  dom  dom  H ) )
2524adantr 467 . . . . . . . . 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 2450 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
27 eqidd 2451 . . . . . . . . . . . 12  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  J  =  dom  dom  J )
2815, 27sscfn1 15715 . . . . . . . . . . 11  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  Fn  ( dom  dom  J  X.  dom  dom  J ) )
2928, 24, 15ssc1 15719 . . . . . . . . . 10  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  J  C_ 
dom  dom  H )
3029sselda 3431 . . . . . . . . 9  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  ->  x  e.  dom  dom  H
)
314, 23, 25, 26, 30subcid 15745 . . . . . . . 8  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  -> 
( ( Id `  C ) `  x
)  =  ( ( Id `  D ) `
 x ) )
3231eleq1d 2512 . . . . . . 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 2823 . . . . . 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 6298 . . . . . . . 8  |-  ( D  |`cat 
J )  =  ( ( C  |`cat  H )  |`cat  J )
356adantr 467 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  C  e.  Cat )
36 dmexg 6721 . . . . . . . . . . 11  |-  ( H  e.  (Subcat `  C
)  ->  dom  H  e. 
_V )
37 dmexg 6721 . . . . . . . . . . 11  |-  ( dom 
H  e.  _V  ->  dom 
dom  H  e.  _V )
3836, 37syl 17 . . . . . . . . . 10  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  e.  _V )
3938adantr 467 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  H  e.  _V )
4035, 24, 28, 39, 29rescabs 15731 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( ( C  |`cat  H )  |`cat  J )  =  ( C  |`cat  J
) )
4134, 40syl5req 2497 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( C  |`cat  J )  =  ( D  |`cat 
J ) )
4241eleq1d 2512 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( ( C  |`cat  J )  e.  Cat  <->  ( D  |`cat  J )  e.  Cat ) )
4322, 33, 423anbi123d 1338 . . . . 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 2450 . . . . . 6  |-  ( C  |`cat 
J )  =  ( C  |`cat  J )
4517, 26, 44, 35, 28issubc3 15747 . . . . 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 2450 . . . . . 6  |-  ( Id
`  D )  =  ( Id `  D
)
47 eqid 2450 . . . . . 6  |-  ( D  |`cat 
J )  =  ( D  |`cat  J )
484, 7subccat 15746 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  D  e.  Cat )
4948adantr 467 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  D  e.  Cat )
502, 46, 47, 49, 28issubc3 15747 . . . . 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 290 . . . 4  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  e.  (Subcat `  D )  <->  J  e.  (Subcat `  C
) ) )
5251pm5.32da 646 . . 3  |-  ( H  e.  (Subcat `  C
)  ->  ( ( J  C_cat  H  /\  J  e.  (Subcat `  D )
)  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  C ) ) ) )
5314, 52bitrd 257 . 2  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  C )
) ) )
54 ancom 452 . 2  |-  ( ( J  C_cat  H  /\  J  e.  (Subcat `  C )
)  <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H )
)
5553, 54syl6bb 265 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 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   A.wral 2736   _Vcvv 3044   class class class wbr 4401    X. cxp 4831   dom cdm 4833    Fn wfn 5576   ` cfv 5581  (class class class)co 6288   Basecbs 15114   Catccat 15563   Idccid 15564   Hom f chomf 15565    C_cat cssc 15705    |`cat cresc 15706  Subcatcsubc 15707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-hom 15207  df-cco 15208  df-cat 15567  df-cid 15568  df-homf 15569  df-ssc 15708  df-resc 15709  df-subc 15710
This theorem is referenced by:  fldhmsubc  40073  fldhmsubcALTV  40092
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