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Theorem resssetc 17127
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 5042 . . . . . . 7 (𝜑𝑉 ∈ V)
54adantr 474 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑉 ∈ V)
6 eqid 2778 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
7 simprl 761 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑥𝑉)
8 simprr 763 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑦𝑉)
91, 5, 6, 7, 8setchom 17115 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑦𝑚 𝑥))
10 resssetc.c . . . . . 6 𝐶 = (SetCat‘𝑈)
112adantr 474 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑈𝑊)
12 eqid 2778 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
133adantr 474 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑉𝑈)
1413, 7sseldd 3822 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑥𝑈)
1513, 8sseldd 3822 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑦𝑈)
1610, 11, 12, 14, 15setchom 17115 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑦𝑚 𝑥))
17 eqid 2778 . . . . . . . 8 (𝐶s 𝑉) = (𝐶s 𝑉)
1817, 12resshom 16464 . . . . . . 7 (𝑉 ∈ V → (Hom ‘𝐶) = (Hom ‘(𝐶s 𝑉)))
194, 18syl 17 . . . . . 6 (𝜑 → (Hom ‘𝐶) = (Hom ‘(𝐶s 𝑉)))
2019oveqdr 6950 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘(𝐶s 𝑉))𝑦))
219, 16, 203eqtr2rd 2821 . . . 4 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
2221ralrimivva 3153 . . 3 (𝜑 → ∀𝑥𝑉𝑦𝑉 (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
23 eqid 2778 . . . 4 (Hom ‘(𝐶s 𝑉)) = (Hom ‘(𝐶s 𝑉))
2410, 2setcbas 17113 . . . . . 6 (𝜑𝑈 = (Base‘𝐶))
253, 24sseqtrd 3860 . . . . 5 (𝜑𝑉 ⊆ (Base‘𝐶))
26 eqid 2778 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2717, 26ressbas2 16327 . . . . 5 (𝑉 ⊆ (Base‘𝐶) → 𝑉 = (Base‘(𝐶s 𝑉)))
2825, 27syl 17 . . . 4 (𝜑𝑉 = (Base‘(𝐶s 𝑉)))
291, 4setcbas 17113 . . . 4 (𝜑𝑉 = (Base‘𝐷))
3023, 6, 28, 29homfeq 16739 . . 3 (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ↔ ∀𝑥𝑉𝑦𝑉 (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
3122, 30mpbird 249 . 2 (𝜑 → (Homf ‘(𝐶s 𝑉)) = (Homf𝐷))
324ad2antrr 716 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉 ∈ V)
33 eqid 2778 . . . . . . . 8 (comp‘𝐷) = (comp‘𝐷)
34 simplr1 1232 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥𝑉)
35 simplr2 1234 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦𝑉)
36 simplr3 1236 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧𝑉)
37 simprl 761 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
381, 32, 6, 34, 35elsetchom 17116 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ↔ 𝑓:𝑥𝑦))
3937, 38mpbid 224 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓:𝑥𝑦)
40 simprr 763 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
411, 32, 6, 35, 36elsetchom 17116 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↔ 𝑔:𝑦𝑧))
4240, 41mpbid 224 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔:𝑦𝑧)
431, 32, 33, 34, 35, 36, 39, 42setcco 17118 . . . . . . 7 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔𝑓))
442ad2antrr 716 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑈𝑊)
45 eqid 2778 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
463ad2antrr 716 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉𝑈)
4746, 34sseldd 3822 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥𝑈)
4846, 35sseldd 3822 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦𝑈)
4946, 36sseldd 3822 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧𝑈)
5010, 44, 45, 47, 48, 49, 39, 42setcco 17118 . . . . . . 7 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔𝑓))
5117, 45ressco 16465 . . . . . . . . . . 11 (𝑉 ∈ V → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
524, 51syl 17 . . . . . . . . . 10 (𝜑 → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
5352ad2antrr 716 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
5453oveqd 6939 . . . . . . . 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 6939 . . . . . . 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 2820 . . . . . 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 3153 . . . . 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 3154 . . . 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 2778 . . . . 5 (comp‘(𝐶s 𝑉)) = (comp‘(𝐶s 𝑉))
6031eqcomd 2784 . . . . 5 (𝜑 → (Homf𝐷) = (Homf ‘(𝐶s 𝑉)))
6133, 59, 6, 29, 28, 60comfeq 16751 . . . 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 249 . . 3 (𝜑 → (compf𝐷) = (compf‘(𝐶s 𝑉)))
6362eqcomd 2784 . 2 (𝜑 → (compf‘(𝐶s 𝑉)) = (compf𝐷))
6431, 63jca 507 1 (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ∧ (compf‘(𝐶s 𝑉)) = (compf𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090  Vcvv 3398  wss 3792  cop 4404  ccom 5359  wf 6131  cfv 6135  (class class class)co 6922  𝑚 cmap 8140  Basecbs 16255  s cress 16256  Hom chom 16349  compcco 16350  Homf chomf 16712  compfccomf 16713  SetCatcsetc 17110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-uz 11993  df-fz 12644  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-hom 16362  df-cco 16363  df-homf 16716  df-comf 16717  df-setc 17111
This theorem is referenced by:  funcsetcres2  17128
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