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Theorem subsubc 17115
Description: A subcategory of a subcategory is a subcategory. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypothesis
Ref Expression
subsubc.d 𝐷 = (𝐶cat 𝐻)
Assertion
Ref Expression
subsubc (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻)))

Proof of Theorem subsubc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . 6 (𝐽 ∈ (Subcat‘𝐷) → 𝐽 ∈ (Subcat‘𝐷))
2 eqid 2825 . . . . . 6 (Homf𝐷) = (Homf𝐷)
31, 2subcssc 17102 . . . . 5 (𝐽 ∈ (Subcat‘𝐷) → 𝐽cat (Homf𝐷))
4 subsubc.d . . . . . . 7 𝐷 = (𝐶cat 𝐻)
5 eqid 2825 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
6 subcrcl 17078 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
7 id 22 . . . . . . . 8 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 ∈ (Subcat‘𝐶))
8 eqidd 2826 . . . . . . . 8 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 = dom dom 𝐻)
97, 8subcfn 17103 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
107, 9, 5subcss1 17104 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ⊆ (Base‘𝐶))
114, 5, 6, 9, 10reschomf 17093 . . . . . 6 (𝐻 ∈ (Subcat‘𝐶) → 𝐻 = (Homf𝐷))
1211breq2d 5074 . . . . 5 (𝐻 ∈ (Subcat‘𝐶) → (𝐽cat 𝐻𝐽cat (Homf𝐷)))
133, 12syl5ibr 247 . . . 4 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) → 𝐽cat 𝐻))
1413pm4.71rd 563 . . 3 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐷))))
15 simpr 485 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat 𝐻)
16 simpl 483 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻 ∈ (Subcat‘𝐶))
17 eqid 2825 . . . . . . . . 9 (Homf𝐶) = (Homf𝐶)
1816, 17subcssc 17102 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻cat (Homf𝐶))
19 ssctr 17087 . . . . . . . 8 ((𝐽cat 𝐻𝐻cat (Homf𝐶)) → 𝐽cat (Homf𝐶))
2015, 18, 19syl2anc 584 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat (Homf𝐶))
2112biimpa 477 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽cat (Homf𝐷))
2220, 212thd 266 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽cat (Homf𝐶) ↔ 𝐽cat (Homf𝐷)))
2316adantr 481 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 ∈ (Subcat‘𝐶))
249adantr 481 . . . . . . . . . 10 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
2524adantr 481 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝐻 Fn (dom dom 𝐻 × dom dom 𝐻))
26 eqid 2825 . . . . . . . . 9 (Id‘𝐶) = (Id‘𝐶)
27 eqidd 2826 . . . . . . . . . . . 12 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐽 = dom dom 𝐽)
2815, 27sscfn1 17079 . . . . . . . . . . 11 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
2928, 24, 15ssc1 17083 . . . . . . . . . 10 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐽 ⊆ dom dom 𝐻)
3029sselda 3970 . . . . . . . . 9 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → 𝑥 ∈ dom dom 𝐻)
314, 23, 25, 26, 30subcid 17109 . . . . . . . 8 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → ((Id‘𝐶)‘𝑥) = ((Id‘𝐷)‘𝑥))
3231eleq1d 2901 . . . . . . 7 (((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) ∧ 𝑥 ∈ dom dom 𝐽) → (((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
3332ralbidva 3200 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥)))
344oveq1i 7161 . . . . . . . 8 (𝐷cat 𝐽) = ((𝐶cat 𝐻) ↾cat 𝐽)
356adantr 481 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐶 ∈ Cat)
36 dmexg 7604 . . . . . . . . . . 11 (𝐻 ∈ (Subcat‘𝐶) → dom 𝐻 ∈ V)
3736dmexd 7606 . . . . . . . . . 10 (𝐻 ∈ (Subcat‘𝐶) → dom dom 𝐻 ∈ V)
3837adantr 481 . . . . . . . . 9 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → dom dom 𝐻 ∈ V)
3935, 24, 28, 38, 29rescabs 17095 . . . . . . . 8 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → ((𝐶cat 𝐻) ↾cat 𝐽) = (𝐶cat 𝐽))
4034, 39syl5req 2873 . . . . . . 7 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐶cat 𝐽) = (𝐷cat 𝐽))
4140eleq1d 2901 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → ((𝐶cat 𝐽) ∈ Cat ↔ (𝐷cat 𝐽) ∈ Cat))
4222, 33, 413anbi123d 1429 . . . . 5 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → ((𝐽cat (Homf𝐶) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐶cat 𝐽) ∈ Cat) ↔ (𝐽cat (Homf𝐷) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐷cat 𝐽) ∈ Cat)))
43 eqid 2825 . . . . . 6 (𝐶cat 𝐽) = (𝐶cat 𝐽)
4417, 26, 43, 35, 28issubc3 17111 . . . . 5 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽cat (Homf𝐶) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐶)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐶cat 𝐽) ∈ Cat)))
45 eqid 2825 . . . . . 6 (Id‘𝐷) = (Id‘𝐷)
46 eqid 2825 . . . . . 6 (𝐷cat 𝐽) = (𝐷cat 𝐽)
474, 7subccat 17110 . . . . . . 7 (𝐻 ∈ (Subcat‘𝐶) → 𝐷 ∈ Cat)
4847adantr 481 . . . . . 6 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → 𝐷 ∈ Cat)
492, 45, 46, 48, 28issubc3 17111 . . . . 5 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽cat (Homf𝐷) ∧ ∀𝑥 ∈ dom dom 𝐽((Id‘𝐷)‘𝑥) ∈ (𝑥𝐽𝑥) ∧ (𝐷cat 𝐽) ∈ Cat)))
5042, 44, 493bitr4rd 313 . . . 4 ((𝐻 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻) → (𝐽 ∈ (Subcat‘𝐷) ↔ 𝐽 ∈ (Subcat‘𝐶)))
5150pm5.32da 579 . . 3 (𝐻 ∈ (Subcat‘𝐶) → ((𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐷)) ↔ (𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐶))))
5214, 51bitrd 280 . 2 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽cat 𝐻𝐽 ∈ (Subcat‘𝐶))))
5352biancomd 464 1 (𝐻 ∈ (Subcat‘𝐶) → (𝐽 ∈ (Subcat‘𝐷) ↔ (𝐽 ∈ (Subcat‘𝐶) ∧ 𝐽cat 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3142  Vcvv 3499   class class class wbr 5062   × cxp 5551  dom cdm 5553   Fn wfn 6346  cfv 6351  (class class class)co 7151  Basecbs 16475  Catccat 16927  Idccid 16928  Homf chomf 16929  cat cssc 17069  cat cresc 17070  Subcatcsubc 17071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-pm 8402  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-hom 16581  df-cco 16582  df-cat 16931  df-cid 16932  df-homf 16933  df-ssc 17072  df-resc 17073  df-subc 17074
This theorem is referenced by:  fldhmsubc  44189  fldhmsubcALTV  44207
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