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Theorem resssetc 16943
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 4957 . . . . . . 7 (𝜑𝑉 ∈ V)
54adantr 472 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑉 ∈ V)
6 eqid 2760 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
7 simprl 811 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑥𝑉)
8 simprr 813 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑦𝑉)
91, 5, 6, 7, 8setchom 16931 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑦𝑚 𝑥))
10 resssetc.c . . . . . 6 𝐶 = (SetCat‘𝑈)
112adantr 472 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑈𝑊)
12 eqid 2760 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
133adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑉𝑈)
1413, 7sseldd 3745 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑥𝑈)
1513, 8sseldd 3745 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑦𝑈)
1610, 11, 12, 14, 15setchom 16931 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑦𝑚 𝑥))
17 eqid 2760 . . . . . . . 8 (𝐶s 𝑉) = (𝐶s 𝑉)
1817, 12resshom 16280 . . . . . . 7 (𝑉 ∈ V → (Hom ‘𝐶) = (Hom ‘(𝐶s 𝑉)))
194, 18syl 17 . . . . . 6 (𝜑 → (Hom ‘𝐶) = (Hom ‘(𝐶s 𝑉)))
2019oveqdr 6837 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘(𝐶s 𝑉))𝑦))
219, 16, 203eqtr2rd 2801 . . . 4 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
2221ralrimivva 3109 . . 3 (𝜑 → ∀𝑥𝑉𝑦𝑉 (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
23 eqid 2760 . . . 4 (Hom ‘(𝐶s 𝑉)) = (Hom ‘(𝐶s 𝑉))
2410, 2setcbas 16929 . . . . . 6 (𝜑𝑈 = (Base‘𝐶))
253, 24sseqtrd 3782 . . . . 5 (𝜑𝑉 ⊆ (Base‘𝐶))
26 eqid 2760 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2717, 26ressbas2 16133 . . . . 5 (𝑉 ⊆ (Base‘𝐶) → 𝑉 = (Base‘(𝐶s 𝑉)))
2825, 27syl 17 . . . 4 (𝜑𝑉 = (Base‘(𝐶s 𝑉)))
291, 4setcbas 16929 . . . 4 (𝜑𝑉 = (Base‘𝐷))
3023, 6, 28, 29homfeq 16555 . . 3 (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ↔ ∀𝑥𝑉𝑦𝑉 (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
3122, 30mpbird 247 . 2 (𝜑 → (Homf ‘(𝐶s 𝑉)) = (Homf𝐷))
324ad2antrr 764 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉 ∈ V)
33 eqid 2760 . . . . . . . 8 (comp‘𝐷) = (comp‘𝐷)
34 simplr1 1261 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥𝑉)
35 simplr2 1263 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦𝑉)
36 simplr3 1265 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧𝑉)
37 simprl 811 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
381, 32, 6, 34, 35elsetchom 16932 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ↔ 𝑓:𝑥𝑦))
3937, 38mpbid 222 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓:𝑥𝑦)
40 simprr 813 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
411, 32, 6, 35, 36elsetchom 16932 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↔ 𝑔:𝑦𝑧))
4240, 41mpbid 222 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔:𝑦𝑧)
431, 32, 33, 34, 35, 36, 39, 42setcco 16934 . . . . . . 7 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔𝑓))
442ad2antrr 764 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑈𝑊)
45 eqid 2760 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
463ad2antrr 764 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉𝑈)
4746, 34sseldd 3745 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥𝑈)
4846, 35sseldd 3745 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦𝑈)
4946, 36sseldd 3745 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧𝑈)
5010, 44, 45, 47, 48, 49, 39, 42setcco 16934 . . . . . . 7 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔𝑓))
5117, 45ressco 16281 . . . . . . . . . . 11 (𝑉 ∈ V → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
524, 51syl 17 . . . . . . . . . 10 (𝜑 → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
5352ad2antrr 764 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
5453oveqd 6830 . . . . . . . 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 6830 . . . . . . 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 2800 . . . . . 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 3109 . . . . 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 3110 . . . 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 2760 . . . . 5 (comp‘(𝐶s 𝑉)) = (comp‘(𝐶s 𝑉))
6031eqcomd 2766 . . . . 5 (𝜑 → (Homf𝐷) = (Homf ‘(𝐶s 𝑉)))
6133, 59, 6, 29, 28, 60comfeq 16567 . . . 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 247 . . 3 (𝜑 → (compf𝐷) = (compf‘(𝐶s 𝑉)))
6362eqcomd 2766 . 2 (𝜑 → (compf‘(𝐶s 𝑉)) = (compf𝐷))
6431, 63jca 555 1 (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ∧ (compf‘(𝐶s 𝑉)) = (compf𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  wss 3715  cop 4327  ccom 5270  wf 6045  cfv 6049  (class class class)co 6813  𝑚 cmap 8023  Basecbs 16059  s cress 16060  Hom chom 16154  compcco 16155  Homf chomf 16528  compfccomf 16529  SetCatcsetc 16926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-hom 16168  df-cco 16169  df-homf 16532  df-comf 16533  df-setc 16927
This theorem is referenced by:  funcsetcres2  16944
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