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Theorem funcres2 17162
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 17126 . . 3 Rel (𝐶 Func (𝐷cat 𝑅))
21a1i 11 . 2 (𝑅 ∈ (Subcat‘𝐷) → Rel (𝐶 Func (𝐷cat 𝑅)))
3 simpr 487 . . . . 5 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔)
4 eqid 2821 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2821 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
6 simpl 485 . . . . . 6 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑅 ∈ (Subcat‘𝐷))
7 eqidd 2822 . . . . . . 7 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → dom dom 𝑅 = dom dom 𝑅)
86, 7subcfn 17105 . . . . . 6 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑅 Fn (dom dom 𝑅 × dom dom 𝑅))
9 eqid 2821 . . . . . . . 8 (Base‘(𝐷cat 𝑅)) = (Base‘(𝐷cat 𝑅))
104, 9, 3funcf1 17130 . . . . . . 7 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘(𝐷cat 𝑅)))
11 eqid 2821 . . . . . . . . 9 (𝐷cat 𝑅) = (𝐷cat 𝑅)
12 eqid 2821 . . . . . . . . 9 (Base‘𝐷) = (Base‘𝐷)
13 subcrcl 17080 . . . . . . . . . 10 (𝑅 ∈ (Subcat‘𝐷) → 𝐷 ∈ Cat)
1413adantr 483 . . . . . . . . 9 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝐷 ∈ Cat)
156, 8, 12subcss1 17106 . . . . . . . . 9 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → dom dom 𝑅 ⊆ (Base‘𝐷))
1611, 12, 14, 8, 15rescbas 17093 . . . . . . . 8 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → dom dom 𝑅 = (Base‘(𝐷cat 𝑅)))
1716feq3d 6496 . . . . . . 7 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → (𝑓:(Base‘𝐶)⟶dom dom 𝑅𝑓:(Base‘𝐶)⟶(Base‘(𝐷cat 𝑅))))
1810, 17mpbird 259 . . . . . 6 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑓:(Base‘𝐶)⟶dom dom 𝑅)
19 eqid 2821 . . . . . . . 8 (Hom ‘(𝐷cat 𝑅)) = (Hom ‘(𝐷cat 𝑅))
20 simplr 767 . . . . . . . 8 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔)
21 simprl 769 . . . . . . . 8 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
22 simprr 771 . . . . . . . 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 17132 . . . . . . 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 17094 . . . . . . . . . 10 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑅 = (Hom ‘(𝐷cat 𝑅)))
2524adantr 483 . . . . . . . . 9 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑅 = (Hom ‘(𝐷cat 𝑅)))
2625oveqd 7167 . . . . . . . 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 6496 . . . . . . 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 259 . . . . . 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 17161 . . . . 5 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → (𝑓(𝐶 Func 𝐷)𝑔𝑓(𝐶 Func (𝐷cat 𝑅))𝑔))
303, 29mpbird 259 . . . 4 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
3130ex 415 . . 3 (𝑅 ∈ (Subcat‘𝐷) → (𝑓(𝐶 Func (𝐷cat 𝑅))𝑔𝑓(𝐶 Func 𝐷)𝑔))
32 df-br 5060 . . 3 (𝑓(𝐶 Func (𝐷cat 𝑅))𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func (𝐷cat 𝑅)))
33 df-br 5060 . . 3 (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
3431, 32, 333imtr3g 297 . 2 (𝑅 ∈ (Subcat‘𝐷) → (⟨𝑓, 𝑔⟩ ∈ (𝐶 Func (𝐷cat 𝑅)) → ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷)))
352, 34relssdv 5656 1 (𝑅 ∈ (Subcat‘𝐷) → (𝐶 Func (𝐷cat 𝑅)) ⊆ (𝐶 Func 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  wss 3936  cop 4567   class class class wbr 5059  dom cdm 5550  Rel wrel 5555  wf 6346  cfv 6350  (class class class)co 7150  Basecbs 16477  Hom chom 16570  Catccat 16929  cat cresc 17072  Subcatcsubc 17073   Func cfunc 17118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-map 8402  df-pm 8403  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-hom 16583  df-cco 16584  df-cat 16933  df-cid 16934  df-homf 16935  df-ssc 17074  df-resc 17075  df-subc 17076  df-func 17122
This theorem is referenced by:  fthres2  17196  ressffth  17202  funcsetcres2  17347
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