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Theorem resssetc 16508
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 4725 . . . . . . 7 (𝜑𝑉 ∈ V)
54adantr 479 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑉 ∈ V)
6 eqid 2606 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
7 simprl 789 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑥𝑉)
8 simprr 791 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑦𝑉)
91, 5, 6, 7, 8setchom 16496 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑦𝑚 𝑥))
10 resssetc.c . . . . . 6 𝐶 = (SetCat‘𝑈)
112adantr 479 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑈𝑊)
12 eqid 2606 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
133adantr 479 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑉𝑈)
1413, 7sseldd 3565 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑥𝑈)
1513, 8sseldd 3565 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑦𝑈)
1610, 11, 12, 14, 15setchom 16496 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑦𝑚 𝑥))
17 eqid 2606 . . . . . . . 8 (𝐶s 𝑉) = (𝐶s 𝑉)
1817, 12resshom 15844 . . . . . . 7 (𝑉 ∈ V → (Hom ‘𝐶) = (Hom ‘(𝐶s 𝑉)))
194, 18syl 17 . . . . . 6 (𝜑 → (Hom ‘𝐶) = (Hom ‘(𝐶s 𝑉)))
2019oveqdr 6548 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘(𝐶s 𝑉))𝑦))
219, 16, 203eqtr2rd 2647 . . . 4 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
2221ralrimivva 2950 . . 3 (𝜑 → ∀𝑥𝑉𝑦𝑉 (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
23 eqid 2606 . . . 4 (Hom ‘(𝐶s 𝑉)) = (Hom ‘(𝐶s 𝑉))
2410, 2setcbas 16494 . . . . . 6 (𝜑𝑈 = (Base‘𝐶))
253, 24sseqtrd 3600 . . . . 5 (𝜑𝑉 ⊆ (Base‘𝐶))
26 eqid 2606 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2717, 26ressbas2 15701 . . . . 5 (𝑉 ⊆ (Base‘𝐶) → 𝑉 = (Base‘(𝐶s 𝑉)))
2825, 27syl 17 . . . 4 (𝜑𝑉 = (Base‘(𝐶s 𝑉)))
291, 4setcbas 16494 . . . 4 (𝜑𝑉 = (Base‘𝐷))
3023, 6, 28, 29homfeq 16120 . . 3 (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ↔ ∀𝑥𝑉𝑦𝑉 (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
3122, 30mpbird 245 . 2 (𝜑 → (Homf ‘(𝐶s 𝑉)) = (Homf𝐷))
324ad2antrr 757 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉 ∈ V)
33 eqid 2606 . . . . . . . 8 (comp‘𝐷) = (comp‘𝐷)
34 simplr1 1095 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥𝑉)
35 simplr2 1096 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦𝑉)
36 simplr3 1097 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧𝑉)
37 simprl 789 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
381, 32, 6, 34, 35elsetchom 16497 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ↔ 𝑓:𝑥𝑦))
3937, 38mpbid 220 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓:𝑥𝑦)
40 simprr 791 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
411, 32, 6, 35, 36elsetchom 16497 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧) ↔ 𝑔:𝑦𝑧))
4240, 41mpbid 220 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔:𝑦𝑧)
431, 32, 33, 34, 35, 36, 39, 42setcco 16499 . . . . . . 7 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔𝑓))
442ad2antrr 757 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑈𝑊)
45 eqid 2606 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
463ad2antrr 757 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉𝑈)
4746, 34sseldd 3565 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥𝑈)
4846, 35sseldd 3565 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦𝑈)
4946, 36sseldd 3565 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧𝑈)
5010, 44, 45, 47, 48, 49, 39, 42setcco 16499 . . . . . . 7 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔𝑓))
5117, 45ressco 15845 . . . . . . . . . . 11 (𝑉 ∈ V → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
524, 51syl 17 . . . . . . . . . 10 (𝜑 → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
5352ad2antrr 757 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
5453oveqd 6541 . . . . . . . 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 6541 . . . . . . 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 2646 . . . . . 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 2950 . . . . 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 2951 . . . 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 2606 . . . . 5 (comp‘(𝐶s 𝑉)) = (comp‘(𝐶s 𝑉))
6031eqcomd 2612 . . . . 5 (𝜑 → (Homf𝐷) = (Homf ‘(𝐶s 𝑉)))
6133, 59, 6, 29, 28, 60comfeq 16132 . . . 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 245 . . 3 (𝜑 → (compf𝐷) = (compf‘(𝐶s 𝑉)))
6362eqcomd 2612 . 2 (𝜑 → (compf‘(𝐶s 𝑉)) = (compf𝐷))
6431, 63jca 552 1 (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ∧ (compf‘(𝐶s 𝑉)) = (compf𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2892  Vcvv 3169  wss 3536  cop 4127  ccom 5029  wf 5783  cfv 5787  (class class class)co 6524  𝑚 cmap 7718  Basecbs 15638  s cress 15639  Hom chom 15722  compcco 15723  Homf chomf 16093  compfccomf 16094  SetCatcsetc 16491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-map 7720  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-2 10923  df-3 10924  df-4 10925  df-5 10926  df-6 10927  df-7 10928  df-8 10929  df-9 10930  df-n0 11137  df-z 11208  df-dec 11323  df-uz 11517  df-fz 12150  df-struct 15640  df-ndx 15641  df-slot 15642  df-base 15643  df-sets 15644  df-ress 15645  df-hom 15736  df-cco 15737  df-homf 16097  df-comf 16098  df-setc 16492
This theorem is referenced by:  funcsetcres2  16509
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