Theorem isfull 16393

Description: Value of the set of full functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
isfull.b	⊢ 𝐵 = (Base‘𝐶)
isfull.j	⊢ 𝐽 = (Hom ‘𝐷)

Assertion

Ref	Expression
isfull	⊢ (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐽,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦

Proof of Theorem isfull

Dummy variables 𝑐 𝑑 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fullfunc 16389	. . 3 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
2	1	ssbri 4627	. 2 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
3		df-br 4584	. . . . . . 7 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4		funcrcl 16346	. . . . . . 7 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
5	3, 4	sylbi 206	. . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6		oveq12 6558	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
7	6	breqd 4594	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑓(𝑐 Func 𝑑)𝑔 ↔ 𝑓(𝐶 Func 𝐷)𝑔))
8		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
9	8	fveq2d 6107	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10		isfull.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10	syl6eqr 2662	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑐) = 𝐵)
12		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
13	12	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Hom ‘𝑑) = (Hom ‘𝐷))
14		isfull.j	. . . . . . . . . . . . . 14 ⊢ 𝐽 = (Hom ‘𝐷)
15	13, 14	syl6eqr 2662	. . . . . . . . . . . . 13 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Hom ‘𝑑) = 𝐽)
16	15	oveqd 6566	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))
17	16	eqeq2d 2620	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)) ↔ ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦))))
18	11, 17	raleqbidv 3129	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦))))
19	11, 18	raleqbidv 3129	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦))))
20	7, 19	anbi12d 743	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦))) ↔ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))))
21	20	opabbidv 4648	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → {⟨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 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))})
22		df-full 16387	. . . . . . 7 ⊢ 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)))})
23		ovex 6577	. . . . . . . 8 ⊢ (𝐶 Func 𝐷) ∈ V
24		simpl 472	. . . . . . . . . 10 ⊢ ((𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦))) → 𝑓(𝐶 Func 𝐷)𝑔)
25	24	ssopab2i 4928	. . . . . . . . 9 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))} ⊆ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔}
26		opabss 4646	. . . . . . . . 9 ⊢ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔} ⊆ (𝐶 Func 𝐷)
27	25, 26	sstri 3577	. . . . . . . 8 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))} ⊆ (𝐶 Func 𝐷)
28	23, 27	ssexi 4731	. . . . . . 7 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))} ∈ V
29	21, 22, 28	ovmpt2a 6689	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))})
30	5, 29	syl 17	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 Full 𝐷) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))})
31	30	breqd 4594	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ 𝐹{⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))}𝐺))
32		relfunc 16345	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
33		brrelex12 5079	. . . . . 6 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
34	32, 33	mpan 702	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
35		breq12 4588	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓(𝐶 Func 𝐷)𝑔 ↔ 𝐹(𝐶 Func 𝐷)𝐺))
36		simpr 476	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
37	36	oveqd 6566	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
38	37	rneqd 5274	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ran (𝑥𝑔𝑦) = ran (𝑥𝐺𝑦))
39		simpl 472	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
40	39	fveq1d 6105	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑥) = (𝐹‘𝑥))
41	39	fveq1d 6105	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑦) = (𝐹‘𝑦))
42	40, 41	oveq12d 6567	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
43	38, 42	eqeq12d 2625	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ↔ ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
44	43	2ralbidv 2972	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
45	35, 44	anbi12d 743	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦))) ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))))
46		eqid 2610	. . . . . 6 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))}
47	45, 46	brabga 4914	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹{⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))}𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))))
48	34, 47	syl 17	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹{⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))}𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))))
49	31, 48	bitrd 267	. . 3 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))))
50	49	bianabs 920	. 2 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
51	2, 50	biadan2 672	1 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⟨cop 4131   class class class wbr 4583  {copab 4642  ran crn 5039  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  Catccat 16148   Func cfunc 16337   Full cful 16385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-func 16341  df-full 16387

This theorem is referenced by:  isfull2  16394  fullpropd  16403  fulloppc  16405  fullres2c  16422