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Theorem resssetc 18114
Description: The restriction of the category of sets to a subset is the category of sets in the subset. Thus, the SetCat‘𝑈 categories for different 𝑈 are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resssetc.c 𝐶 = (SetCat‘𝑈)
resssetc.d 𝐷 = (SetCat‘𝑉)
resssetc.1 (𝜑𝑈𝑊)
resssetc.2 (𝜑𝑉𝑈)
Assertion
Ref Expression
resssetc (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ∧ (compf‘(𝐶s 𝑉)) = (compf𝐷)))

Proof of Theorem resssetc
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resssetc.d . . . . . 6 𝐷 = (SetCat‘𝑉)
2 resssetc.1 . . . . . . . 8 (𝜑𝑈𝑊)
3 resssetc.2 . . . . . . . 8 (𝜑𝑉𝑈)
42, 3ssexd 5329 . . . . . . 7 (𝜑𝑉 ∈ V)
54adantr 479 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑉 ∈ V)
6 eqid 2726 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
7 simprl 769 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑥𝑉)
8 simprr 771 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑦𝑉)
91, 5, 6, 7, 8setchom 18102 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑦m 𝑥))
10 resssetc.c . . . . . 6 𝐶 = (SetCat‘𝑈)
112adantr 479 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑈𝑊)
12 eqid 2726 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
133adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑉𝑈)
1413, 7sseldd 3980 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑥𝑈)
1513, 8sseldd 3980 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑦𝑈)
1610, 11, 12, 14, 15setchom 18102 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑦m 𝑥))
17 eqid 2726 . . . . . . . 8 (𝐶s 𝑉) = (𝐶s 𝑉)
1817, 12resshom 17433 . . . . . . 7 (𝑉 ∈ V → (Hom ‘𝐶) = (Hom ‘(𝐶s 𝑉)))
194, 18syl 17 . . . . . 6 (𝜑 → (Hom ‘𝐶) = (Hom ‘(𝐶s 𝑉)))
2019oveqdr 7452 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘(𝐶s 𝑉))𝑦))
219, 16, 203eqtr2rd 2773 . . . 4 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
2221ralrimivva 3191 . . 3 (𝜑 → ∀𝑥𝑉𝑦𝑉 (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
23 eqid 2726 . . . 4 (Hom ‘(𝐶s 𝑉)) = (Hom ‘(𝐶s 𝑉))
2410, 2setcbas 18100 . . . . . 6 (𝜑𝑈 = (Base‘𝐶))
253, 24sseqtrd 4020 . . . . 5 (𝜑𝑉 ⊆ (Base‘𝐶))
26 eqid 2726 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2717, 26ressbas2 17251 . . . . 5 (𝑉 ⊆ (Base‘𝐶) → 𝑉 = (Base‘(𝐶s 𝑉)))
2825, 27syl 17 . . . 4 (𝜑𝑉 = (Base‘(𝐶s 𝑉)))
291, 4setcbas 18100 . . . 4 (𝜑𝑉 = (Base‘𝐷))
3023, 6, 28, 29homfeq 17707 . . 3 (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ↔ ∀𝑥𝑉𝑦𝑉 (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
3122, 30mpbird 256 . 2 (𝜑 → (Homf ‘(𝐶s 𝑉)) = (Homf𝐷))
324ad2antrr 724 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉 ∈ V)
33 eqid 2726 . . . . . . . 8 (comp‘𝐷) = (comp‘𝐷)
34 simplr1 1212 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥𝑉)
35 simplr2 1213 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦𝑉)
36 simplr3 1214 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧𝑉)
37 simprl 769 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
381, 32, 6, 34, 35elsetchom 18103 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ↔ 𝑓:𝑥𝑦))
3937, 38mpbid 231 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓:𝑥𝑦)
40 simprr 771 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
411, 32, 6, 35, 36elsetchom 18103 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↔ 𝑔:𝑦𝑧))
4240, 41mpbid 231 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔:𝑦𝑧)
431, 32, 33, 34, 35, 36, 39, 42setcco 18105 . . . . . . 7 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔𝑓))
442ad2antrr 724 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑈𝑊)
45 eqid 2726 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
463ad2antrr 724 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉𝑈)
4746, 34sseldd 3980 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥𝑈)
4846, 35sseldd 3980 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦𝑈)
4946, 36sseldd 3980 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧𝑈)
5010, 44, 45, 47, 48, 49, 39, 42setcco 18105 . . . . . . 7 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔𝑓))
5117, 45ressco 17434 . . . . . . . . . . 11 (𝑉 ∈ V → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
524, 51syl 17 . . . . . . . . . 10 (𝜑 → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
5352ad2antrr 724 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
5453oveqd 7441 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧) = (⟨𝑥, 𝑦⟩(comp‘(𝐶s 𝑉))𝑧))
5554oveqd 7441 . . . . . . 7 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶s 𝑉))𝑧)𝑓))
5643, 50, 553eqtr2d 2772 . . . . . 6 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶s 𝑉))𝑧)𝑓))
5756ralrimivva 3191 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶s 𝑉))𝑧)𝑓))
5857ralrimivvva 3194 . . . 4 (𝜑 → ∀𝑥𝑉𝑦𝑉𝑧𝑉𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶s 𝑉))𝑧)𝑓))
59 eqid 2726 . . . . 5 (comp‘(𝐶s 𝑉)) = (comp‘(𝐶s 𝑉))
6031eqcomd 2732 . . . . 5 (𝜑 → (Homf𝐷) = (Homf ‘(𝐶s 𝑉)))
6133, 59, 6, 29, 28, 60comfeq 17719 . . . 4 (𝜑 → ((compf𝐷) = (compf‘(𝐶s 𝑉)) ↔ ∀𝑥𝑉𝑦𝑉𝑧𝑉𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶s 𝑉))𝑧)𝑓)))
6258, 61mpbird 256 . . 3 (𝜑 → (compf𝐷) = (compf‘(𝐶s 𝑉)))
6362eqcomd 2732 . 2 (𝜑 → (compf‘(𝐶s 𝑉)) = (compf𝐷))
6431, 63jca 510 1 (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ∧ (compf‘(𝐶s 𝑉)) = (compf𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  Vcvv 3462  wss 3947  cop 4639  ccom 5686  wf 6550  cfv 6554  (class class class)co 7424  m cmap 8855  Basecbs 17213  s cress 17242  Hom chom 17277  compcco 17278  Homf chomf 17679  compfccomf 17680  SetCatcsetc 18097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12611  df-dec 12730  df-uz 12875  df-fz 13539  df-struct 17149  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-hom 17290  df-cco 17291  df-homf 17683  df-comf 17684  df-setc 18098
This theorem is referenced by:  funcsetcres2  18115
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