Theorem fullfunc 17864

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 7363	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐 Full 𝑑) = (𝐶 Full 𝑑))
2		oveq1 7363	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐 Func 𝑑) = (𝐶 Func 𝑑))
3	1, 2	sseq12d 3950	. . 3 ⊢ (𝑐 = 𝐶 → ((𝑐 Full 𝑑) ⊆ (𝑐 Func 𝑑) ↔ (𝐶 Full 𝑑) ⊆ (𝐶 Func 𝑑)))
4		oveq2 7364	. . . 4 ⊢ (𝑑 = 𝐷 → (𝐶 Full 𝑑) = (𝐶 Full 𝐷))
5		oveq2 7364	. . . 4 ⊢ (𝑑 = 𝐷 → (𝐶 Func 𝑑) = (𝐶 Func 𝐷))
6	4, 5	sseq12d 3950	. . 3 ⊢ (𝑑 = 𝐷 → ((𝐶 Full 𝑑) ⊆ (𝐶 Func 𝑑) ↔ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)))
7		ovex 7389	. . . . . 6 ⊢ (𝑐 Func 𝑑) ∈ V
8		simpl 482	. . . . . . . 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 5494	. . . . . . 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 5138	. . . . . . 7 ⊢ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝑐 Func 𝑑)𝑔} ⊆ (𝑐 Func 𝑑)
11	9, 10	sstri 3926	. . . . . 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 5252	. . . . 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 17862	. . . . . 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 7503	. . . . 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 1453	. . . 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 3961	. . 3 ⊢ ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Full 𝑑) ⊆ (𝑐 Func 𝑑))
17	3, 6, 16	vtocl2ga 3519	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷))
18	13	mpondm0 7596	. . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) = ∅)
19		0ss 4330	. . 3 ⊢ ∅ ⊆ (𝐶 Func 𝐷)
20	18, 19	eqsstrdi 3961	. 2 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷))
21	17, 20	pm2.61i 182	1 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3049  Vcvv 3427   ⊆ wss 3885  ∅c0 4263   class class class wbr 5074  {copab 5136  ran crn 5621  ‘cfv 6487  (class class class)co 7356  Basecbs 17168  Hom chom 17220  Catccat 17619   Func cfunc 17810   Full cful 17860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6443  df-fun 6489  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-full 17862

This theorem is referenced by:  relfull  17866  isfull  17868  fulloppc  17880  cofull  17892  catcisolem  18066  catciso  18067  imasubc  49614  imasubc2  49615  idfullsubc  49624  fulloppf  49626  uptrlem1  49673  uptrlem2  49674  uptrlem3  49675  uptra  49678  uptrar  49679  uobeqw  49682  uobeq  49683  uptr2  49684  uptr2a  49685  fucoppcfunc  49875  fullthinc2  49914  thincciso  49916  fulltermc2  49975  termfucterm  50007  uobeqterm  50009