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Theorem funcres2 16917
Description: A functor into a restricted category is also a functor into the whole category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
funcres2 (𝑅 ∈ (Subcat‘𝐷) → (𝐶 Func (𝐷cat 𝑅)) ⊆ (𝐶 Func 𝐷))

Proof of Theorem funcres2
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 16881 . . 3 Rel (𝐶 Func (𝐷cat 𝑅))
21a1i 11 . 2 (𝑅 ∈ (Subcat‘𝐷) → Rel (𝐶 Func (𝐷cat 𝑅)))
3 simpr 479 . . . . 5 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔)
4 eqid 2825 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2825 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
6 simpl 476 . . . . . 6 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑅 ∈ (Subcat‘𝐷))
7 eqidd 2826 . . . . . . 7 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → dom dom 𝑅 = dom dom 𝑅)
86, 7subcfn 16860 . . . . . 6 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑅 Fn (dom dom 𝑅 × dom dom 𝑅))
9 eqid 2825 . . . . . . . 8 (Base‘(𝐷cat 𝑅)) = (Base‘(𝐷cat 𝑅))
104, 9, 3funcf1 16885 . . . . . . 7 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘(𝐷cat 𝑅)))
11 eqid 2825 . . . . . . . . 9 (𝐷cat 𝑅) = (𝐷cat 𝑅)
12 eqid 2825 . . . . . . . . 9 (Base‘𝐷) = (Base‘𝐷)
13 subcrcl 16835 . . . . . . . . . 10 (𝑅 ∈ (Subcat‘𝐷) → 𝐷 ∈ Cat)
1413adantr 474 . . . . . . . . 9 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝐷 ∈ Cat)
156, 8, 12subcss1 16861 . . . . . . . . 9 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → dom dom 𝑅 ⊆ (Base‘𝐷))
1611, 12, 14, 8, 15rescbas 16848 . . . . . . . 8 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → dom dom 𝑅 = (Base‘(𝐷cat 𝑅)))
1716feq3d 6269 . . . . . . 7 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → (𝑓:(Base‘𝐶)⟶dom dom 𝑅𝑓:(Base‘𝐶)⟶(Base‘(𝐷cat 𝑅))))
1810, 17mpbird 249 . . . . . 6 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑓:(Base‘𝐶)⟶dom dom 𝑅)
19 eqid 2825 . . . . . . . 8 (Hom ‘(𝐷cat 𝑅)) = (Hom ‘(𝐷cat 𝑅))
20 simplr 785 . . . . . . . 8 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔)
21 simprl 787 . . . . . . . 8 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
22 simprr 789 . . . . . . . 8 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
234, 5, 19, 20, 21, 22funcf2 16887 . . . . . . 7 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝑔𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝑓𝑥)(Hom ‘(𝐷cat 𝑅))(𝑓𝑦)))
2411, 12, 14, 8, 15reschom 16849 . . . . . . . . . 10 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑅 = (Hom ‘(𝐷cat 𝑅)))
2524adantr 474 . . . . . . . . 9 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑅 = (Hom ‘(𝐷cat 𝑅)))
2625oveqd 6927 . . . . . . . 8 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝑓𝑥)𝑅(𝑓𝑦)) = ((𝑓𝑥)(Hom ‘(𝐷cat 𝑅))(𝑓𝑦)))
2726feq3d 6269 . . . . . . 7 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((𝑥𝑔𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝑓𝑥)𝑅(𝑓𝑦)) ↔ (𝑥𝑔𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝑓𝑥)(Hom ‘(𝐷cat 𝑅))(𝑓𝑦))))
2823, 27mpbird 249 . . . . . 6 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥𝑔𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶((𝑓𝑥)𝑅(𝑓𝑦)))
294, 5, 6, 8, 18, 28funcres2b 16916 . . . . 5 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → (𝑓(𝐶 Func 𝐷)𝑔𝑓(𝐶 Func (𝐷cat 𝑅))𝑔))
303, 29mpbird 249 . . . 4 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
3130ex 403 . . 3 (𝑅 ∈ (Subcat‘𝐷) → (𝑓(𝐶 Func (𝐷cat 𝑅))𝑔𝑓(𝐶 Func 𝐷)𝑔))
32 df-br 4876 . . 3 (𝑓(𝐶 Func (𝐷cat 𝑅))𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func (𝐷cat 𝑅)))
33 df-br 4876 . . 3 (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
3431, 32, 333imtr3g 287 . 2 (𝑅 ∈ (Subcat‘𝐷) → (⟨𝑓, 𝑔⟩ ∈ (𝐶 Func (𝐷cat 𝑅)) → ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷)))
352, 34relssdv 5450 1 (𝑅 ∈ (Subcat‘𝐷) → (𝐶 Func (𝐷cat 𝑅)) ⊆ (𝐶 Func 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wss 3798  cop 4405   class class class wbr 4875  dom cdm 5346  Rel wrel 5351  wf 6123  cfv 6127  (class class class)co 6910  Basecbs 16229  Hom chom 16323  Catccat 16684  cat cresc 16827  Subcatcsubc 16828   Func cfunc 16873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-er 8014  df-map 8129  df-pm 8130  df-ixp 8182  df-en 8229  df-dom 8230  df-sdom 8231  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-3 11422  df-4 11423  df-5 11424  df-6 11425  df-7 11426  df-8 11427  df-9 11428  df-n0 11626  df-z 11712  df-dec 11829  df-ndx 16232  df-slot 16233  df-base 16235  df-sets 16236  df-ress 16237  df-hom 16336  df-cco 16337  df-cat 16688  df-cid 16689  df-homf 16690  df-ssc 16829  df-resc 16830  df-subc 16831  df-func 16877
This theorem is referenced by:  fthres2  16951  ressffth  16957  funcsetcres2  17102
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