Theorem isfull 17288

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 17284	. . 3 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
2	1	ssbri 5076	. 2 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
3		df-br 5032	. . . . . . 7 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4		funcrcl 17241	. . . . . . 7 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
5	3, 4	sylbi 220	. . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6		oveq12 7182	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
7	6	breqd 5042	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑓(𝑐 Func 𝑑)𝑔 ↔ 𝑓(𝐶 Func 𝐷)𝑔))
8		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
9	8	fveq2d 6681	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10		isfull.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10	eqtr4di 2792	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑐) = 𝐵)
12		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
13	12	fveq2d 6681	. . . . . . . . . . . . . 14 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Hom ‘𝑑) = (Hom ‘𝐷))
14		isfull.j	. . . . . . . . . . . . . 14 ⊢ 𝐽 = (Hom ‘𝐷)
15	13, 14	eqtr4di 2792	. . . . . . . . . . . . 13 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Hom ‘𝑑) = 𝐽)
16	15	oveqd 7190	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))
17	16	eqeq2d 2750	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)) ↔ ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦))))
18	11, 17	raleqbidv 3305	. . . . . . . . . 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 3305	. . . . . . . . 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 634	. . . . . . . 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 5097	. . . . . . 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 17282	. . . . . . 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 7206	. . . . . . . 8 ⊢ (𝐶 Func 𝐷) ∈ V
24		simpl 486	. . . . . . . . . 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 5406	. . . . . . . . 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 5095	. . . . . . . . 9 ⊢ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔} ⊆ (𝐶 Func 𝐷)
27	25, 26	sstri 3887	. . . . . . . 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 5191	. . . . . . 7 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))} ∈ V
29	21, 22, 28	ovmpoa 7323	. . . . . 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 5042	. . . 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 17240	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
33	32	brrelex12i 5579	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
34		breq12 5036	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓(𝐶 Func 𝐷)𝑔 ↔ 𝐹(𝐶 Func 𝐷)𝐺))
35		simpr 488	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
36	35	oveqd 7190	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
37	36	rneqd 5782	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ran (𝑥𝑔𝑦) = ran (𝑥𝐺𝑦))
38		simpl 486	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
39	38	fveq1d 6679	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑥) = (𝐹‘𝑥))
40	38	fveq1d 6679	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑦) = (𝐹‘𝑦))
41	39, 40	oveq12d 7191	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
42	37, 41	eqeq12d 2755	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ↔ ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
43	42	2ralbidv 3112	. . . . . . 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))))
44	34, 43	anbi12d 634	. . . . . 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)))))
45		eqid 2739	. . . . . 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)))}
46	44, 45	brabga 5390	. . . . 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)))))
47	33, 46	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)))))
48	31, 47	bitrd 282	. . 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)))))
49	48	bianabs 545	. 2 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
50	2, 49	biadanii 822	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 209   ∧ wa 399   = wceq 1542   ∈ wcel 2114  ∀wral 3054  Vcvv 3399  ⟨cop 4523   class class class wbr 5031  {copab 5093  ran crn 5527  ‘cfv 6340  (class class class)co 7173  Basecbs 16589  Hom chom 16682  Catccat 17041   Func cfunc 17232   Full cful 17280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-un 7482

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-1st 7717  df-2nd 7718  df-func 17236  df-full 17282

This theorem is referenced by:  isfull2  17289  fullpropd  17298  fulloppc  17300  fullres2c  17317