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Theorem subsubc 13727
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 19 . . . . . 6  |-  ( J  e.  (Subcat `  D
)  ->  J  e.  (Subcat `  D ) )
2 eqid 2283 . . . . . 6  |-  (  Homf  `  D )  =  (  Homf 
`  D )
31, 2subcssc 13714 . . . . 5  |-  ( J  e.  (Subcat `  D
)  ->  J  C_cat  (  Homf  `  D ) )
4 subsubc.d . . . . . . 7  |-  D  =  ( C  |`cat  H )
5 eqid 2283 . . . . . . 7  |-  ( Base `  C )  =  (
Base `  C )
6 subcrcl 13693 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  Cat )
7 id 19 . . . . . . . 8  |-  ( H  e.  (Subcat `  C
)  ->  H  e.  (Subcat `  C ) )
8 eqidd 2284 . . . . . . . 8  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  =  dom  dom  H )
97, 8subcfn 13715 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  H  Fn  ( dom  dom  H  X.  dom  dom  H ) )
107, 9, 5subcss1 13716 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  C_  ( Base `  C
) )
114, 5, 6, 9, 10reschomf 13708 . . . . . 6  |-  ( H  e.  (Subcat `  C
)  ->  H  =  (  Homf 
`  D ) )
1211breq2d 4035 . . . . 5  |-  ( H  e.  (Subcat `  C
)  ->  ( J  C_cat  H  <-> 
J  C_cat  (  Homf 
`  D ) ) )
133, 12syl5ibr 212 . . . 4  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  ->  J  C_cat  H ) )
1413pm4.71rd 616 . . 3  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  D )
) ) )
15 simpr 447 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  H )
16 simpl 443 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  e.  (Subcat `  C ) )
17 eqid 2283 . . . . . . . . 9  |-  (  Homf  `  C )  =  (  Homf 
`  C )
1816, 17subcssc 13714 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  C_cat  (  Homf  `  C ) )
19 ssctr 13702 . . . . . . . 8  |-  ( ( J  C_cat  H  /\  H  C_cat  (  Homf  `  C ) )  ->  J  C_cat  (  Homf 
`  C ) )
2015, 18, 19syl2anc 642 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  (  Homf  `  C ) )
2112biimpa 470 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  C_cat  (  Homf  `  D ) )
2220, 212thd 231 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  C_cat  (  Homf 
`  C )  <->  J  C_cat  (  Homf  `  D ) ) )
2316adantr 451 . . . . . . . . 9  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  ->  H  e.  (Subcat `  C
) )
249adantr 451 . . . . . . . . . 10  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  H  Fn  ( dom  dom  H  X.  dom  dom  H ) )
2524adantr 451 . . . . . . . . 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 2283 . . . . . . . . 9  |-  ( Id
`  C )  =  ( Id `  C
)
27 eqidd 2284 . . . . . . . . . . . 12  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  J  =  dom  dom  J )
2815, 27sscfn1 13694 . . . . . . . . . . 11  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  J  Fn  ( dom  dom  J  X.  dom  dom  J ) )
2928, 24, 15ssc1 13698 . . . . . . . . . 10  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  J  C_ 
dom  dom  H )
3029sselda 3180 . . . . . . . . 9  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  ->  x  e.  dom  dom  H
)
314, 23, 25, 26, 30subcid 13721 . . . . . . . 8  |-  ( ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  /\  x  e. 
dom  dom  J )  -> 
( ( Id `  C ) `  x
)  =  ( ( Id `  D ) `
 x ) )
3231eleq1d 2349 . . . . . . 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 2559 . . . . . 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 5868 . . . . . . . 8  |-  ( D  |`cat 
J )  =  ( ( C  |`cat  H )  |`cat  J )
356adantr 451 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  C  e.  Cat )
36 dmexg 4939 . . . . . . . . . . 11  |-  ( H  e.  (Subcat `  C
)  ->  dom  H  e. 
_V )
37 dmexg 4939 . . . . . . . . . . 11  |-  ( dom 
H  e.  _V  ->  dom 
dom  H  e.  _V )
3836, 37syl 15 . . . . . . . . . 10  |-  ( H  e.  (Subcat `  C
)  ->  dom  dom  H  e.  _V )
3938adantr 451 . . . . . . . . 9  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  dom  dom  H  e.  _V )
4035, 24, 28, 39, 29rescabs 13710 . . . . . . . 8  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( ( C  |`cat  H )  |`cat  J )  =  ( C  |`cat  J
) )
4134, 40syl5req 2328 . . . . . . 7  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( C  |`cat  J )  =  ( D  |`cat 
J ) )
4241eleq1d 2349 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( ( C  |`cat  J )  e.  Cat  <->  ( D  |`cat  J )  e.  Cat ) )
4322, 33, 423anbi123d 1252 . . . . 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 2283 . . . . . 6  |-  ( C  |`cat 
J )  =  ( C  |`cat  J )
4517, 26, 44, 35, 28issubc3 13723 . . . . 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 2283 . . . . . 6  |-  ( Id
`  D )  =  ( Id `  D
)
47 eqid 2283 . . . . . 6  |-  ( D  |`cat 
J )  =  ( D  |`cat  J )
484, 7subccat 13722 . . . . . . 7  |-  ( H  e.  (Subcat `  C
)  ->  D  e.  Cat )
4948adantr 451 . . . . . 6  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  D  e.  Cat )
502, 46, 47, 49, 28issubc3 13723 . . . . 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 277 . . . 4  |-  ( ( H  e.  (Subcat `  C )  /\  J  C_cat  H )  ->  ( J  e.  (Subcat `  D )  <->  J  e.  (Subcat `  C
) ) )
5251pm5.32da 622 . . 3  |-  ( H  e.  (Subcat `  C
)  ->  ( ( J  C_cat  H  /\  J  e.  (Subcat `  D )
)  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  C ) ) ) )
5314, 52bitrd 244 . 2  |-  ( H  e.  (Subcat `  C
)  ->  ( J  e.  (Subcat `  D )  <->  ( J  C_cat  H  /\  J  e.  (Subcat `  C )
) ) )
54 ancom 437 . 2  |-  ( ( J  C_cat  H  /\  J  e.  (Subcat `  C )
)  <->  ( J  e.  (Subcat `  C )  /\  J  C_cat  H )
)
5553, 54syl6bb 252 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   class class class wbr 4023    X. cxp 4687   dom cdm 4689    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   Basecbs 13148   Catccat 13566   Idccid 13567    Homf chomf 13568    C_cat cssc 13684    |`cat cresc 13685  Subcatcsubc 13686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-hom 13232  df-cco 13233  df-cat 13570  df-cid 13571  df-homf 13572  df-ssc 13687  df-resc 13688  df-subc 13689
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