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Theorem funcres2 16605
 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 16569 . . 3 Rel (𝐶 Func (𝐷cat 𝑅))
21a1i 11 . 2 (𝑅 ∈ (Subcat‘𝐷) → Rel (𝐶 Func (𝐷cat 𝑅)))
3 simpr 476 . . . . 5 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔)
4 eqid 2651 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
5 eqid 2651 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
6 simpl 472 . . . . . 6 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑅 ∈ (Subcat‘𝐷))
7 eqidd 2652 . . . . . . 7 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → dom dom 𝑅 = dom dom 𝑅)
86, 7subcfn 16548 . . . . . 6 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑅 Fn (dom dom 𝑅 × dom dom 𝑅))
9 eqid 2651 . . . . . . . 8 (Base‘(𝐷cat 𝑅)) = (Base‘(𝐷cat 𝑅))
104, 9, 3funcf1 16573 . . . . . . 7 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑓:(Base‘𝐶)⟶(Base‘(𝐷cat 𝑅)))
11 eqid 2651 . . . . . . . . 9 (𝐷cat 𝑅) = (𝐷cat 𝑅)
12 eqid 2651 . . . . . . . . 9 (Base‘𝐷) = (Base‘𝐷)
13 subcrcl 16523 . . . . . . . . . 10 (𝑅 ∈ (Subcat‘𝐷) → 𝐷 ∈ Cat)
1413adantr 480 . . . . . . . . 9 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝐷 ∈ Cat)
156, 8, 12subcss1 16549 . . . . . . . . 9 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → dom dom 𝑅 ⊆ (Base‘𝐷))
1611, 12, 14, 8, 15rescbas 16536 . . . . . . . 8 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → dom dom 𝑅 = (Base‘(𝐷cat 𝑅)))
1716feq3d 6070 . . . . . . 7 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → (𝑓:(Base‘𝐶)⟶dom dom 𝑅𝑓:(Base‘𝐶)⟶(Base‘(𝐷cat 𝑅))))
1810, 17mpbird 247 . . . . . 6 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑓:(Base‘𝐶)⟶dom dom 𝑅)
19 eqid 2651 . . . . . . . 8 (Hom ‘(𝐷cat 𝑅)) = (Hom ‘(𝐷cat 𝑅))
20 simplr 807 . . . . . . . 8 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔)
21 simprl 809 . . . . . . . 8 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
22 simprr 811 . . . . . . . 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 16575 . . . . . . 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 16537 . . . . . . . . . 10 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑅 = (Hom ‘(𝐷cat 𝑅)))
2524adantr 480 . . . . . . . . 9 (((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑅 = (Hom ‘(𝐷cat 𝑅)))
2625oveqd 6707 . . . . . . . 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 6070 . . . . . . 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 247 . . . . . 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 16604 . . . . 5 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → (𝑓(𝐶 Func 𝐷)𝑔𝑓(𝐶 Func (𝐷cat 𝑅))𝑔))
303, 29mpbird 247 . . . 4 ((𝑅 ∈ (Subcat‘𝐷) ∧ 𝑓(𝐶 Func (𝐷cat 𝑅))𝑔) → 𝑓(𝐶 Func 𝐷)𝑔)
3130ex 449 . . 3 (𝑅 ∈ (Subcat‘𝐷) → (𝑓(𝐶 Func (𝐷cat 𝑅))𝑔𝑓(𝐶 Func 𝐷)𝑔))
32 df-br 4686 . . 3 (𝑓(𝐶 Func (𝐷cat 𝑅))𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func (𝐷cat 𝑅)))
33 df-br 4686 . . 3 (𝑓(𝐶 Func 𝐷)𝑔 ↔ ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷))
3431, 32, 333imtr3g 284 . 2 (𝑅 ∈ (Subcat‘𝐷) → (⟨𝑓, 𝑔⟩ ∈ (𝐶 Func (𝐷cat 𝑅)) → ⟨𝑓, 𝑔⟩ ∈ (𝐶 Func 𝐷)))
352, 34relssdv 5246 1 (𝑅 ∈ (Subcat‘𝐷) → (𝐶 Func (𝐷cat 𝑅)) ⊆ (𝐶 Func 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ⊆ wss 3607  ⟨cop 4216   class class class wbr 4685  dom cdm 5143  Rel wrel 5148  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  Hom chom 15999  Catccat 16372   ↾cat cresc 16515  Subcatcsubc 16516   Func cfunc 16561 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-hom 16013  df-cco 16014  df-cat 16376  df-cid 16377  df-homf 16378  df-ssc 16517  df-resc 16518  df-subc 16519  df-func 16565 This theorem is referenced by:  fthres2  16639  ressffth  16645  funcsetcres2  16790
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