Theorem fthfunc 18002

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 7423	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐 Faith 𝑑) = (𝐶 Faith 𝑑))
2		oveq1 7423	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐 Func 𝑑) = (𝐶 Func 𝑑))
3	1, 2	sseq12d 3967	. . 3 ⊢ (𝑐 = 𝐶 → ((𝑐 Faith 𝑑) ⊆ (𝑐 Func 𝑑) ↔ (𝐶 Faith 𝑑) ⊆ (𝐶 Func 𝑑)))
4		oveq2 7424	. . . 4 ⊢ (𝑑 = 𝐷 → (𝐶 Faith 𝑑) = (𝐶 Faith 𝐷))
5		oveq2 7424	. . . 4 ⊢ (𝑑 = 𝐷 → (𝐶 Func 𝑑) = (𝐶 Func 𝐷))
6	4, 5	sseq12d 3967	. . 3 ⊢ (𝑑 = 𝐷 → ((𝐶 Faith 𝑑) ⊆ (𝐶 Func 𝑑) ↔ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)))
7		ovex 7449	. . . . . 6 ⊢ (𝑐 Func 𝑑) ∈ V
8		simpl 488	. . . . . . . 8 ⊢ ((𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦)) → 𝑓(𝑐 Func 𝑑)𝑔)
9	8	ssopab2i 5533	. . . . . . 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 5173	. . . . . . 7 ⊢ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝑐 Func 𝑑)𝑔} ⊆ (𝑐 Func 𝑑)
11	9, 10	sstri 3943	. . . . . 6 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))} ⊆ (𝑐 Func 𝑑)
12	7, 11	ssexi 5291	. . . . 5 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))} ∈ V
13		df-fth 18000	. . . . . 6 ⊢ Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))})
14	13	ovmpt4g 7563	. . . . 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 1479	. . . 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 3978	. . 3 ⊢ ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Faith 𝑑) ⊆ (𝑐 Func 𝑑))
17	3, 6, 16	vtocl2ga 3540	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷))
18	13	mpondm0 7657	. . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) = ∅)
19		0ss 4353	. . 3 ⊢ ∅ ⊆ (𝐶 Func 𝐷)
20	18, 19	eqsstrdi 3978	. 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 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   ⊆ wss 3902  ∅c0 4282   class class class wbr 5107  {copab 5171  ^◡ccnv 5658  Fun wfun 6531  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  Catccat 17756   Func cfunc 17947   Faith cfth 17998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-fth 18000

This theorem is used by:  relfth  18004  isfth  18009  fthoppc  18018  fthsect  18020  fthinv  18021  fthmon  18022  fthepi  18023  ffthiso  18024  cofth  18030  inclfusubc  18036  fthcomf  50070  fthoppf  50077