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Theorem resssetc 17881
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 5262 . . . . . . 7 (𝜑𝑉 ∈ V)
54adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑉 ∈ V)
6 eqid 2736 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
7 simprl 768 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑥𝑉)
8 simprr 770 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑦𝑉)
91, 5, 6, 7, 8setchom 17869 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑦m 𝑥))
10 resssetc.c . . . . . 6 𝐶 = (SetCat‘𝑈)
112adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑈𝑊)
12 eqid 2736 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
133adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑉𝑈)
1413, 7sseldd 3931 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑥𝑈)
1513, 8sseldd 3931 . . . . . 6 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → 𝑦𝑈)
1610, 11, 12, 14, 15setchom 17869 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑦m 𝑥))
17 eqid 2736 . . . . . . . 8 (𝐶s 𝑉) = (𝐶s 𝑉)
1817, 12resshom 17203 . . . . . . 7 (𝑉 ∈ V → (Hom ‘𝐶) = (Hom ‘(𝐶s 𝑉)))
194, 18syl 17 . . . . . 6 (𝜑 → (Hom ‘𝐶) = (Hom ‘(𝐶s 𝑉)))
2019oveqdr 7344 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘(𝐶s 𝑉))𝑦))
219, 16, 203eqtr2rd 2783 . . . 4 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
2221ralrimivva 3193 . . 3 (𝜑 → ∀𝑥𝑉𝑦𝑉 (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
23 eqid 2736 . . . 4 (Hom ‘(𝐶s 𝑉)) = (Hom ‘(𝐶s 𝑉))
2410, 2setcbas 17867 . . . . . 6 (𝜑𝑈 = (Base‘𝐶))
253, 24sseqtrd 3970 . . . . 5 (𝜑𝑉 ⊆ (Base‘𝐶))
26 eqid 2736 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
2717, 26ressbas2 17023 . . . . 5 (𝑉 ⊆ (Base‘𝐶) → 𝑉 = (Base‘(𝐶s 𝑉)))
2825, 27syl 17 . . . 4 (𝜑𝑉 = (Base‘(𝐶s 𝑉)))
291, 4setcbas 17867 . . . 4 (𝜑𝑉 = (Base‘𝐷))
3023, 6, 28, 29homfeq 17477 . . 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 723 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉 ∈ V)
33 eqid 2736 . . . . . . . 8 (comp‘𝐷) = (comp‘𝐷)
34 simplr1 1214 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥𝑉)
35 simplr2 1215 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦𝑉)
36 simplr3 1216 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧𝑉)
37 simprl 768 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
381, 32, 6, 34, 35elsetchom 17870 . . . . . . . . 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 770 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
411, 32, 6, 35, 36elsetchom 17870 . . . . . . . . 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 17872 . . . . . . 7 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔𝑓))
442ad2antrr 723 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑈𝑊)
45 eqid 2736 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
463ad2antrr 723 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉𝑈)
4746, 34sseldd 3931 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥𝑈)
4846, 35sseldd 3931 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦𝑈)
4946, 36sseldd 3931 . . . . . . . 8 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧𝑈)
5010, 44, 45, 47, 48, 49, 39, 42setcco 17872 . . . . . . 7 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔𝑓))
5117, 45ressco 17204 . . . . . . . . . . 11 (𝑉 ∈ V → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
524, 51syl 17 . . . . . . . . . 10 (𝜑 → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
5352ad2antrr 723 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
5453oveqd 7333 . . . . . . . 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 7333 . . . . . . 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 2782 . . . . . 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 3193 . . . . 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 3196 . . . 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 2736 . . . . 5 (comp‘(𝐶s 𝑉)) = (comp‘(𝐶s 𝑉))
6031eqcomd 2742 . . . . 5 (𝜑 → (Homf𝐷) = (Homf ‘(𝐶s 𝑉)))
6133, 59, 6, 29, 28, 60comfeq 17489 . . . 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 2742 . 2 (𝜑 → (compf‘(𝐶s 𝑉)) = (compf𝐷))
6431, 63jca 512 1 (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ∧ (compf‘(𝐶s 𝑉)) = (compf𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1540  wcel 2105  wral 3061  Vcvv 3440  wss 3896  cop 4576  ccom 5611  wf 6461  cfv 6465  (class class class)co 7316  m cmap 8664  Basecbs 16986  s cress 17015  Hom chom 17047  compcco 17048  Homf chomf 17449  compfccomf 17450  SetCatcsetc 17864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629  ax-cnex 11006  ax-resscn 11007  ax-1cn 11008  ax-icn 11009  ax-addcl 11010  ax-addrcl 11011  ax-mulcl 11012  ax-mulrcl 11013  ax-mulcom 11014  ax-addass 11015  ax-mulass 11016  ax-distr 11017  ax-i2m1 11018  ax-1ne0 11019  ax-1rid 11020  ax-rnegex 11021  ax-rrecex 11022  ax-cnre 11023  ax-pre-lttri 11024  ax-pre-lttrn 11025  ax-pre-ltadd 11026  ax-pre-mulgt0 11027
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4850  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5170  df-tr 5204  df-id 5506  df-eprel 5512  df-po 5520  df-so 5521  df-fr 5562  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-pred 6224  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-riota 7273  df-ov 7319  df-oprab 7320  df-mpo 7321  df-om 7759  df-1st 7877  df-2nd 7878  df-frecs 8145  df-wrecs 8176  df-recs 8250  df-rdg 8289  df-1o 8345  df-er 8547  df-map 8666  df-en 8783  df-dom 8784  df-sdom 8785  df-fin 8786  df-pnf 11090  df-mnf 11091  df-xr 11092  df-ltxr 11093  df-le 11094  df-sub 11286  df-neg 11287  df-nn 12053  df-2 12115  df-3 12116  df-4 12117  df-5 12118  df-6 12119  df-7 12120  df-8 12121  df-9 12122  df-n0 12313  df-z 12399  df-dec 12517  df-uz 12662  df-fz 13319  df-struct 16922  df-sets 16939  df-slot 16957  df-ndx 16969  df-base 16987  df-ress 17016  df-hom 17060  df-cco 17061  df-homf 17453  df-comf 17454  df-setc 17865
This theorem is referenced by:  funcsetcres2  17882
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