Theorem fullfunc 17170

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.

Step	Hyp	Ref	Expression
1		oveq1 7157	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐 Full 𝑑) = (𝐶 Full 𝑑))
2		oveq1 7157	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐 Func 𝑑) = (𝐶 Func 𝑑))
3	1, 2	sseq12d 3999	. . 3 ⊢ (𝑐 = 𝐶 → ((𝑐 Full 𝑑) ⊆ (𝑐 Func 𝑑) ↔ (𝐶 Full 𝑑) ⊆ (𝐶 Func 𝑑)))
4		oveq2 7158	. . . 4 ⊢ (𝑑 = 𝐷 → (𝐶 Full 𝑑) = (𝐶 Full 𝐷))
5		oveq2 7158	. . . 4 ⊢ (𝑑 = 𝐷 → (𝐶 Func 𝑑) = (𝐶 Func 𝐷))
6	4, 5	sseq12d 3999	. . 3 ⊢ (𝑑 = 𝐷 → ((𝐶 Full 𝑑) ⊆ (𝐶 Func 𝑑) ↔ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)))
7		ovex 7183	. . . . . 6 ⊢ (𝑐 Func 𝑑) ∈ V
8		simpl 485	. . . . . . . 8 ⊢ ((𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦))) → 𝑓(𝑐 Func 𝑑)𝑔)
9	8	ssopab2i 5429	. . . . . . 7 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)))} ⊆ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝑐 Func 𝑑)𝑔}
10		opabss 5122	. . . . . . 7 ⊢ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝑐 Func 𝑑)𝑔} ⊆ (𝑐 Func 𝑑)
11	9, 10	sstri 3975	. . . . . 6 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)))} ⊆ (𝑐 Func 𝑑)
12	7, 11	ssexi 5218	. . . . 5 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)))} ∈ V
13		df-full 17168	. . . . . 6 ⊢ Full = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)))})
14	13	ovmpt4g 7291	. . . . 5 ⊢ ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat ∧ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)))} ∈ V) → (𝑐 Full 𝑑) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)))})
15	12, 14	mp3an3 1446	. . . 4 ⊢ ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Full 𝑑) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)))})
16	15, 11	eqsstrdi 4020	. . 3 ⊢ ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Full 𝑑) ⊆ (𝑐 Func 𝑑))
17	3, 6, 16	vtocl2ga 3574	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷))
18	13	mpondm0 7380	. . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) = ∅)
19		0ss 4349	. . 3 ⊢ ∅ ⊆ (𝐶 Func 𝐷)
20	18, 19	eqsstrdi 4020	. 2 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷))
21	17, 20	pm2.61i 184	1 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)

Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494   ⊆ wss 3935  ∅c0 4290   class class class wbr 5058  {copab 5120  ran crn 5550  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  Hom chom 16570  Catccat 16929   Func cfunc 17118   Full cful 17166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-full 17168

This theorem is referenced by:  relfull  17172  isfull  17174  fulloppc  17186  cofull  17198  catcisolem  17360  catciso  17361