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Theorem fullfunc 17798
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 7365 . . . 4 (𝑐 = 𝐶 → (𝑐 Full 𝑑) = (𝐶 Full 𝑑))
2 oveq1 7365 . . . 4 (𝑐 = 𝐶 → (𝑐 Func 𝑑) = (𝐶 Func 𝑑))
31, 2sseq12d 3978 . . 3 (𝑐 = 𝐶 → ((𝑐 Full 𝑑) ⊆ (𝑐 Func 𝑑) ↔ (𝐶 Full 𝑑) ⊆ (𝐶 Func 𝑑)))
4 oveq2 7366 . . . 4 (𝑑 = 𝐷 → (𝐶 Full 𝑑) = (𝐶 Full 𝐷))
5 oveq2 7366 . . . 4 (𝑑 = 𝐷 → (𝐶 Func 𝑑) = (𝐶 Func 𝐷))
64, 5sseq12d 3978 . . 3 (𝑑 = 𝐷 → ((𝐶 Full 𝑑) ⊆ (𝐶 Func 𝑑) ↔ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)))
7 ovex 7391 . . . . . 6 (𝑐 Func 𝑑) ∈ V
8 simpl 484 . . . . . . . 8 ((𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦))) → 𝑓(𝑐 Func 𝑑)𝑔)
98ssopab2i 5508 . . . . . . 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 5170 . . . . . . 7 {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝑐 Func 𝑑)𝑔} ⊆ (𝑐 Func 𝑑)
119, 10sstri 3954 . . . . . 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 5280 . . . . 5 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))} ∈ V
13 df-full 17796 . . . . . 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 7503 . . . . 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 1451 . . . 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 3999 . . 3 ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Full 𝑑) ⊆ (𝑐 Func 𝑑))
173, 6, 16vtocl2ga 3534 . 2 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷))
1813mpondm0 7595 . . 3 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) = ∅)
19 0ss 4357 . . 3 ∅ ⊆ (𝐶 Func 𝐷)
2018, 19eqsstrdi 3999 . 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 397   = wceq 1542  wcel 2107  wral 3061  Vcvv 3444  wss 3911  c0 4283   class class class wbr 5106  {copab 5168  ran crn 5635  cfv 6497  (class class class)co 7358  Basecbs 17088  Hom chom 17149  Catccat 17549   Func cfunc 17745   Full cful 17794
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-full 17796
This theorem is referenced by:  relfull  17800  isfull  17802  fulloppc  17814  cofull  17826  catcisolem  18001  catciso  18002  fullthinc2  47153  thincciso  47155
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