Theorem isfull 16955

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 16951	. . 3 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
2	1	ssbri 4931	. 2 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
3		df-br 4887	. . . . . . 7 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4		funcrcl 16908	. . . . . . 7 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
5	3, 4	sylbi 209	. . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6		oveq12 6931	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
7	6	breqd 4897	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑓(𝑐 Func 𝑑)𝑔 ↔ 𝑓(𝐶 Func 𝐷)𝑔))
8		simpl 476	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
9	8	fveq2d 6450	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10		isfull.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10	syl6eqr 2831	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑐) = 𝐵)
12		simpr 479	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
13	12	fveq2d 6450	. . . . . . . . . . . . . 14 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Hom ‘𝑑) = (Hom ‘𝐷))
14		isfull.j	. . . . . . . . . . . . . 14 ⊢ 𝐽 = (Hom ‘𝐷)
15	13, 14	syl6eqr 2831	. . . . . . . . . . . . 13 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Hom ‘𝑑) = 𝐽)
16	15	oveqd 6939	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)))
17	16	eqeq2d 2787	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝑑)(𝑓‘𝑦)) ↔ ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦))))
18	11, 17	raleqbidv 3325	. . . . . . . . . 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 3325	. . . . . . . . 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 624	. . . . . . . 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 4952	. . . . . . 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 16949	. . . . . . 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 6954	. . . . . . . 8 ⊢ (𝐶 Func 𝐷) ∈ V
24		simpl 476	. . . . . . . . . 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 5240	. . . . . . . . 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 4950	. . . . . . . . 9 ⊢ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔} ⊆ (𝐶 Func 𝐷)
27	25, 26	sstri 3829	. . . . . . . 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 5040	. . . . . . 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 7068	. . . . . 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 4897	. . . 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 16907	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
33	32	brrelex12i 5405	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
34		breq12 4891	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓(𝐶 Func 𝐷)𝑔 ↔ 𝐹(𝐶 Func 𝐷)𝐺))
35		simpr 479	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
36	35	oveqd 6939	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
37	36	rneqd 5598	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ran (𝑥𝑔𝑦) = ran (𝑥𝐺𝑦))
38		simpl 476	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑓 = 𝐹)
39	38	fveq1d 6448	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑥) = (𝐹‘𝑥))
40	38	fveq1d 6448	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓‘𝑦) = (𝐹‘𝑦))
41	39, 40	oveq12d 6940	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
42	37, 41	eqeq12d 2792	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) ↔ ran (𝑥𝐺𝑦) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
43	42	2ralbidv 3170	. . . . . . 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 624	. . . . . 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 2777	. . . . . 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 5226	. . . . 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 271	. . 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 537	. 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 813	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 198   ∧ wa 386   = wceq 1601   ∈ wcel 2106  ∀wral 3089  Vcvv 3397  ⟨cop 4403   class class class wbr 4886  {copab 4948  ran crn 5356  ‘cfv 6135  (class class class)co 6922  Basecbs 16255  Hom chom 16349  Catccat 16710   Func cfunc 16899   Full cful 16947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-func 16903  df-full 16949

This theorem is referenced by:  isfull2  16956  fullpropd  16965  fulloppc  16967  fullres2c  16984