Theorem isfth 16574

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

Hypothesis

Ref	Expression
isfth.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
isfth	⊢ (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝐺𝑦)))

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

Proof of Theorem isfth

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

Step	Hyp	Ref	Expression
1		fthfunc 16567	. . 3 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
2	1	ssbri 4697	. 2 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
3		df-br 4654	. . . . . . 7 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4		funcrcl 16523	. . . . . . 7 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
5	3, 4	sylbi 207	. . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6		oveq12 6659	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
7	6	breqd 4664	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑓(𝑐 Func 𝑑)𝑔 ↔ 𝑓(𝐶 Func 𝐷)𝑔))
8		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
9	8	fveq2d 6195	. . . . . . . . . . 11 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10		isfth.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐶)
11	9, 10	syl6eqr 2674	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (Base‘𝑐) = 𝐵)
12	11	raleqdv 3144	. . . . . . . . . 10 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦)))
13	11, 12	raleqbidv 3152	. . . . . . . . 9 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦)))
14	7, 13	anbi12d 747	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦)) ↔ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦))))
15	14	opabbidv 4716	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))} = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦))})
16		df-fth 16565	. . . . . . 7 ⊢ Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))})
17		ovex 6678	. . . . . . . 8 ⊢ (𝐶 Func 𝐷) ∈ V
18		simpl 473	. . . . . . . . . 10 ⊢ ((𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦)) → 𝑓(𝐶 Func 𝐷)𝑔)
19	18	ssopab2i 5003	. . . . . . . . 9 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦))} ⊆ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔}
20		opabss 4714	. . . . . . . . 9 ⊢ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔} ⊆ (𝐶 Func 𝐷)
21	19, 20	sstri 3612	. . . . . . . 8 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦))} ⊆ (𝐶 Func 𝐷)
22	17, 21	ssexi 4803	. . . . . . 7 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦))} ∈ V
23	15, 16, 22	ovmpt2a 6791	. . . . . 6 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦))})
24	5, 23	syl 17	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 Faith 𝐷) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦))})
25	24	breqd 4664	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ 𝐹{⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦))}𝐺))
26		relfunc 16522	. . . . . 6 ⊢ Rel (𝐶 Func 𝐷)
27		brrelex12 5155	. . . . . 6 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
28	26, 27	mpan 706	. . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
29		breq12 4658	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑓(𝐶 Func 𝐷)𝑔 ↔ 𝐹(𝐶 Func 𝐷)𝐺))
30		simpr 477	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
31	30	oveqd 6667	. . . . . . . . . 10 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
32	31	cnveqd 5298	. . . . . . . . 9 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ◡(𝑥𝑔𝑦) = ◡(𝑥𝐺𝑦))
33	32	funeqd 5910	. . . . . . . 8 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (Fun ◡(𝑥𝑔𝑦) ↔ Fun ◡(𝑥𝐺𝑦)))
34	33	2ralbidv 2989	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝐺𝑦)))
35	29, 34	anbi12d 747	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦)) ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝐺𝑦))))
36		eqid 2622	. . . . . 6 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦))} = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦))}
37	35, 36	brabga 4989	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹{⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦))}𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝐺𝑦))))
38	28, 37	syl 17	. . . 4 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹{⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝑔𝑦))}𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝐺𝑦))))
39	25, 38	bitrd 268	. . 3 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝐺𝑦))))
40	39	bianabs 924	. 2 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝐺𝑦)))
41	2, 40	biadan2 674	1 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 Fun ◡(𝑥𝐺𝑦)))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  ⟨cop 4183   class class class wbr 4653  {copab 4712  ^◡ccnv 5113  Rel wrel 5119  Fun wfun 5882  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Catccat 16325   Func cfunc 16514   Faith cfth 16563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-func 16518  df-fth 16565

This theorem is referenced by:  isfth2  16575  fthpropd  16581  fthoppc  16583  fthres2b  16590  fthres2c  16591  fthres2  16592