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Theorem fullfunc 17622
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 7282 . . . 4 (𝑐 = 𝐶 → (𝑐 Full 𝑑) = (𝐶 Full 𝑑))
2 oveq1 7282 . . . 4 (𝑐 = 𝐶 → (𝑐 Func 𝑑) = (𝐶 Func 𝑑))
31, 2sseq12d 3954 . . 3 (𝑐 = 𝐶 → ((𝑐 Full 𝑑) ⊆ (𝑐 Func 𝑑) ↔ (𝐶 Full 𝑑) ⊆ (𝐶 Func 𝑑)))
4 oveq2 7283 . . . 4 (𝑑 = 𝐷 → (𝐶 Full 𝑑) = (𝐶 Full 𝐷))
5 oveq2 7283 . . . 4 (𝑑 = 𝐷 → (𝐶 Func 𝑑) = (𝐶 Func 𝐷))
64, 5sseq12d 3954 . . 3 (𝑑 = 𝐷 → ((𝐶 Full 𝑑) ⊆ (𝐶 Func 𝑑) ↔ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)))
7 ovex 7308 . . . . . 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 5463 . . . . . . 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 5138 . . . . . . 7 {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝑐 Func 𝑑)𝑔} ⊆ (𝑐 Func 𝑑)
119, 10sstri 3930 . . . . . 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 5246 . . . . 5 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))} ∈ V
13 df-full 17620 . . . . . 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 7420 . . . . 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 1449 . . . 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 3975 . . 3 ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Full 𝑑) ⊆ (𝑐 Func 𝑑))
173, 6, 16vtocl2ga 3514 . 2 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷))
1813mpondm0 7510 . . 3 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) = ∅)
19 0ss 4330 . . 3 ∅ ⊆ (𝐶 Func 𝐷)
2018, 19eqsstrdi 3975 . 2 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷))
2117, 20pm2.61i 182 1 (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1539  wcel 2106  wral 3064  Vcvv 3432  wss 3887  c0 4256   class class class wbr 5074  {copab 5136  ran crn 5590  cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973  Catccat 17373   Func cfunc 17569   Full cful 17618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-full 17620
This theorem is referenced by:  relfull  17624  isfull  17626  fulloppc  17638  cofull  17650  catcisolem  17825  catciso  17826  fullthinc2  46328  thincciso  46330
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