Theorem fthfunc 17972

Description: A faithful functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)

Assertion

Ref	Expression
fthfunc	⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)

Proof of Theorem fthfunc

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

Step	Hyp	Ref	Expression
1		oveq1 7419	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐 Faith 𝑑) = (𝐶 Faith 𝑑))
2		oveq1 7419	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐 Func 𝑑) = (𝐶 Func 𝑑))
3	1, 2	sseq12d 3969	. . 3 ⊢ (𝑐 = 𝐶 → ((𝑐 Faith 𝑑) ⊆ (𝑐 Func 𝑑) ↔ (𝐶 Faith 𝑑) ⊆ (𝐶 Func 𝑑)))
4		oveq2 7420	. . . 4 ⊢ (𝑑 = 𝐷 → (𝐶 Faith 𝑑) = (𝐶 Faith 𝐷))
5		oveq2 7420	. . . 4 ⊢ (𝑑 = 𝐷 → (𝐶 Func 𝑑) = (𝐶 Func 𝐷))
6	4, 5	sseq12d 3969	. . 3 ⊢ (𝑑 = 𝐷 → ((𝐶 Faith 𝑑) ⊆ (𝐶 Func 𝑑) ↔ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)))
7		ovex 7445	. . . . . 6 ⊢ (𝑐 Func 𝑑) ∈ V
8		simpl 487	. . . . . . . 8 ⊢ ((𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦)) → 𝑓(𝑐 Func 𝑑)𝑔)
9	8	ssopab2i 5534	. . . . . . 7 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))} ⊆ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝑐 Func 𝑑)𝑔}
10		opabss 5174	. . . . . . 7 ⊢ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝑐 Func 𝑑)𝑔} ⊆ (𝑐 Func 𝑑)
11	9, 10	sstri 3945	. . . . . 6 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))} ⊆ (𝑐 Func 𝑑)
12	7, 11	ssexi 5292	. . . . 5 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))} ∈ V
13		df-fth 17970	. . . . . 6 ⊢ Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))})
14	13	ovmpt4g 7559	. . . . 5 ⊢ ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat ∧ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))} ∈ V) → (𝑐 Faith 𝑑) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))})
15	12, 14	mp3an3 1478	. . . 4 ⊢ ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Faith 𝑑) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))})
16	15, 11	eqsstrdi 3980	. . 3 ⊢ ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Faith 𝑑) ⊆ (𝑐 Func 𝑑))
17	3, 6, 16	vtocl2ga 3541	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷))
18	13	mpondm0 7652	. . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) = ∅)
19		0ss 4356	. . 3 ⊢ ∅ ⊆ (𝐶 Func 𝐷)
20	18, 19	eqsstrdi 3980	. 2 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷))
21	17, 20	pm2.61i 184	1 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  Vcvv 3454   ⊆ wss 3904  ∅c0 4285   class class class wbr 5108  {copab 5172  ^◡ccnv 5659  Fun wfun 6530  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  Catccat 17726   Func cfunc 17917   Faith cfth 17968

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-fth 17970

This theorem is used by:  relfth  17974  isfth  17979  fthoppc  17988  fthsect  17990  fthinv  17991  fthmon  17992  fthepi  17993  ffthiso  17994  cofth  18000  inclfusubc  18006  fthcomf  49963  fthoppf  49970