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Theorem fullfunc 17168
Description: A full functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
Assertion
Ref Expression
fullfunc (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)

Proof of Theorem fullfunc
Dummy variables 𝑐 𝑑 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7158 . . . 4 (𝑐 = 𝐶 → (𝑐 Full 𝑑) = (𝐶 Full 𝑑))
2 oveq1 7158 . . . 4 (𝑐 = 𝐶 → (𝑐 Func 𝑑) = (𝐶 Func 𝑑))
31, 2sseq12d 4003 . . 3 (𝑐 = 𝐶 → ((𝑐 Full 𝑑) ⊆ (𝑐 Func 𝑑) ↔ (𝐶 Full 𝑑) ⊆ (𝐶 Func 𝑑)))
4 oveq2 7159 . . . 4 (𝑑 = 𝐷 → (𝐶 Full 𝑑) = (𝐶 Full 𝐷))
5 oveq2 7159 . . . 4 (𝑑 = 𝐷 → (𝐶 Func 𝑑) = (𝐶 Func 𝐷))
64, 5sseq12d 4003 . . 3 (𝑑 = 𝐷 → ((𝐶 Full 𝑑) ⊆ (𝐶 Func 𝑑) ↔ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)))
7 ovex 7184 . . . . . 6 (𝑐 Func 𝑑) ∈ V
8 simpl 483 . . . . . . . 8 ((𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦))) → 𝑓(𝑐 Func 𝑑)𝑔)
98ssopab2i 5433 . . . . . . 7 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))} ⊆ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝑐 Func 𝑑)𝑔}
10 opabss 5126 . . . . . . 7 {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝑐 Func 𝑑)𝑔} ⊆ (𝑐 Func 𝑑)
119, 10sstri 3979 . . . . . 6 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))} ⊆ (𝑐 Func 𝑑)
127, 11ssexi 5222 . . . . 5 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))} ∈ V
13 df-full 17166 . . . . . 6 Full = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))})
1413ovmpt4g 7290 . . . . 5 ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat ∧ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))} ∈ V) → (𝑐 Full 𝑑) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))})
1512, 14mp3an3 1443 . . . 4 ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Full 𝑑) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))})
1615, 11eqsstrdi 4024 . . 3 ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Full 𝑑) ⊆ (𝑐 Func 𝑑))
173, 6, 16vtocl2ga 3579 . 2 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷))
1813mpondm0 7379 . . 3 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) = ∅)
19 0ss 4353 . . 3 ∅ ⊆ (𝐶 Func 𝐷)
2018, 19eqsstrdi 4024 . 2 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷))
2117, 20pm2.61i 183 1 (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1530  wcel 2106  wral 3142  Vcvv 3499  wss 3939  c0 4294   class class class wbr 5062  {copab 5124  ran crn 5554  cfv 6351  (class class class)co 7151  Basecbs 16475  Hom chom 16568  Catccat 16927   Func cfunc 17116   Full cful 17164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-full 17166
This theorem is referenced by:  relfull  17170  isfull  17172  fulloppc  17184  cofull  17196  catcisolem  17358  catciso  17359
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