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Theorem subsubc 14050
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 20 . . . . . 6  |-  ( J  e.  (Subcat `  D
)  ->  J  e.  (Subcat `  D ) )
2 eqid 2436 . . . . . 6  |-  (  Homf  `  D )  =  (  Homf 
`  D )
31, 2subcssc 14037 . . . . 5  |-  ( J  e.  (Subcat `  D
)  ->  J  C_cat  (  Homf  `  D ) )
4 subsubc.d . . . . . . 7  |-  D  =  ( C  |`cat  H )
5 eqid 2436 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
6 subcrcl 14016 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  Cat )
7 id 20 . . . . . . . 8  |-  ( H  e.  (Subcat `  C
)  ->  H  e.  (Subcat `  C ) )
8 eqidd 2437 . . . . . . . 8  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  =  dom  dom  H )
97, 8subcfn 14038 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  H  Fn  ( dom  dom  H  X.  dom  dom  H ) )
107, 9, 5subcss1 14039 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  C_  ( Base `  C
) )
114, 5, 6, 9, 10reschomf 14031 . . . . . 6  |-  ( H  e.  (Subcat `  C
)  ->  H  =  (  Homf 
`  D ) )
1211breq2d 4224 . . . . 5  |-  ( H  e.  (Subcat `  C
)  ->  ( J  C_cat  H  <-> 
J  C_cat  (  Homf 
`  D ) ) )
133, 12syl5ibr 213 . . . 4  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  ->  J  C_cat  H ) )
1413pm4.71rd 617 . . 3  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  D )
) ) )
15 simpr 448 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  H )
16 simpl 444 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  e.  (Subcat `  C ) )
17 eqid 2436 . . . . . . . . 9  |-  (  Homf  `  C )  =  (  Homf 
`  C )
1816, 17subcssc 14037 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  C_cat  (  Homf  `  C ) )
19 ssctr 14025 . . . . . . . 8  |-  ( ( J  C_cat  H  /\  H  C_cat  (  Homf  `  C ) )  ->  J  C_cat  (  Homf 
`  C ) )
2015, 18, 19syl2anc 643 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  (  Homf  `  C ) )
2112biimpa 471 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  (  Homf  `  D ) )
2220, 212thd 232 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  C_cat  (  Homf 
`  C )  <->  J  C_cat  (  Homf  `  D ) ) )
2316adantr 452 . . . . . . . . 9  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  ->  H  e.  (Subcat `  C
) )
249adantr 452 . . . . . . . . . 10  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  Fn  ( dom  dom  H  X.  dom  dom  H ) )
2524adantr 452 . . . . . . . . 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 2436 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
27 eqidd 2437 . . . . . . . . . . . 12  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  J  =  dom  dom  J )
2815, 27sscfn1 14017 . . . . . . . . . . 11  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  Fn  ( dom  dom  J  X.  dom  dom  J ) )
2928, 24, 15ssc1 14021 . . . . . . . . . 10  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  J  C_ 
dom  dom  H )
3029sselda 3348 . . . . . . . . 9  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  ->  x  e.  dom  dom  H
)
314, 23, 25, 26, 30subcid 14044 . . . . . . . 8  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  -> 
( ( Id `  C ) `  x
)  =  ( ( Id `  D ) `
 x ) )
3231eleq1d 2502 . . . . . . 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 2721 . . . . . 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 6091 . . . . . . . 8  |-  ( D  |`cat 
J )  =  ( ( C  |`cat  H )  |`cat  J )
356adantr 452 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  C  e.  Cat )
36 dmexg 5130 . . . . . . . . . . 11  |-  ( H  e.  (Subcat `  C
)  ->  dom  H  e. 
_V )
37 dmexg 5130 . . . . . . . . . . 11  |-  ( dom 
H  e.  _V  ->  dom 
dom  H  e.  _V )
3836, 37syl 16 . . . . . . . . . 10  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  e.  _V )
3938adantr 452 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  H  e.  _V )
4035, 24, 28, 39, 29rescabs 14033 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( ( C  |`cat  H )  |`cat  J )  =  ( C  |`cat  J
) )
4134, 40syl5req 2481 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( C  |`cat  J )  =  ( D  |`cat 
J ) )
4241eleq1d 2502 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( ( C  |`cat  J )  e.  Cat  <->  ( D  |`cat  J )  e.  Cat ) )
4322, 33, 423anbi123d 1254 . . . . 5  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( ( J  C_cat  (  Homf 
`  C )  /\  A. x  e.  dom  dom  J ( ( Id `  C ) `  x
)  e.  ( x J x )  /\  ( C  |`cat  J )  e.  Cat ) 
<->  ( J  C_cat  (  Homf  `  D )  /\  A. x  e.  dom  dom  J
( ( Id `  D ) `  x
)  e.  ( x J x )  /\  ( D  |`cat  J )  e.  Cat ) ) )
44 eqid 2436 . . . . . 6  |-  ( C  |`cat 
J )  =  ( C  |`cat  J )
4517, 26, 44, 35, 28issubc3 14046 . . . . 5  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  e.  (Subcat `  C )  <->  ( J  C_cat  (  Homf 
`  C )  /\  A. x  e.  dom  dom  J ( ( Id `  C ) `  x
)  e.  ( x J x )  /\  ( C  |`cat  J )  e.  Cat ) ) )
46 eqid 2436 . . . . . 6  |-  ( Id
`  D )  =  ( Id `  D
)
47 eqid 2436 . . . . . 6  |-  ( D  |`cat 
J )  =  ( D  |`cat  J )
484, 7subccat 14045 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  D  e.  Cat )
4948adantr 452 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  D  e.  Cat )
502, 46, 47, 49, 28issubc3 14046 . . . . 5  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  e.  (Subcat `  D )  <->  ( J  C_cat  (  Homf 
`  D )  /\  A. x  e.  dom  dom  J ( ( Id `  D ) `  x
)  e.  ( x J x )  /\  ( D  |`cat  J )  e.  Cat ) ) )
5143, 45, 503bitr4rd 278 . . . 4  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  e.  (Subcat `  D )  <->  J  e.  (Subcat `  C
) ) )
5251pm5.32da 623 . . 3  |-  ( H  e.  (Subcat `  C
)  ->  ( ( J  C_cat  H  /\  J  e.  (Subcat `  D )
)  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  C ) ) ) )
5314, 52bitrd 245 . 2  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  C )
) ) )
54 ancom 438 . 2  |-  ( ( J  C_cat  H  /\  J  e.  (Subcat `  C )
)  <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H )
)
5553, 54syl6bb 253 1  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   class class class wbr 4212    X. cxp 4876   dom cdm 4878    Fn wfn 5449   ` cfv 5454  (class class class)co 6081   Basecbs 13469   Catccat 13889   Idccid 13890    Homf chomf 13891    C_cat cssc 14007    |`cat cresc 14008  Subcatcsubc 14009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-hom 13553  df-cco 13554  df-cat 13893  df-cid 13894  df-homf 13895  df-ssc 14010  df-resc 14011  df-subc 14012
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