Theorem fthfunc 17914

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 7388	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐 Faith 𝑑) = (𝐶 Faith 𝑑))
2		oveq1 7388	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑐 Func 𝑑) = (𝐶 Func 𝑑))
3	1, 2	sseq12d 3960	. . 3 ⊢ (𝑐 = 𝐶 → ((𝑐 Faith 𝑑) ⊆ (𝑐 Func 𝑑) ↔ (𝐶 Faith 𝑑) ⊆ (𝐶 Func 𝑑)))
4		oveq2 7389	. . . 4 ⊢ (𝑑 = 𝐷 → (𝐶 Faith 𝑑) = (𝐶 Faith 𝐷))
5		oveq2 7389	. . . 4 ⊢ (𝑑 = 𝐷 → (𝐶 Func 𝑑) = (𝐶 Func 𝐷))
6	4, 5	sseq12d 3960	. . 3 ⊢ (𝑑 = 𝐷 → ((𝐶 Faith 𝑑) ⊆ (𝐶 Func 𝑑) ↔ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)))
7		ovex 7414	. . . . . 6 ⊢ (𝑐 Func 𝑑) ∈ V
8		simpl 485	. . . . . . . 8 ⊢ ((𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦)) → 𝑓(𝑐 Func 𝑑)𝑔)
9	8	ssopab2i 5510	. . . . . . 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 5154	. . . . . . 7 ⊢ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝑐 Func 𝑑)𝑔} ⊆ (𝑐 Func 𝑑)
11	9, 10	sstri 3936	. . . . . 6 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))} ⊆ (𝑐 Func 𝑑)
12	7, 11	ssexi 5268	. . . . 5 ⊢ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))} ∈ V
13		df-fth 17912	. . . . . 6 ⊢ Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun ◡(𝑥𝑔𝑦))})
14	13	ovmpt4g 7528	. . . . 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 1461	. . . 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 3971	. . 3 ⊢ ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Faith 𝑑) ⊆ (𝑐 Func 𝑑))
17	3, 6, 16	vtocl2ga 3533	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷))
18	13	mpondm0 7621	. . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) = ∅)
19		0ss 4344	. . 3 ⊢ ∅ ⊆ (𝐶 Func 𝐷)
20	18, 19	eqsstrdi 3971	. 2 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷))
21	17, 20	pm2.61i 183	1 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)

Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1550   ∈ wcel 2132  ∀wral 3066  Vcvv 3444   ⊆ wss 3895  ∅c0 4276   class class class wbr 5090  {copab 5152  ^◡ccnv 5635  Fun wfun 6500  ‘cfv 6506  (class class class)co 7381  Basecbs 17217  Catccat 17668   Func cfunc 17859   Faith cfth 17910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-iota 6462  df-fun 6508  df-fv 6514  df-ov 7384  df-oprab 7385  df-mpo 7386  df-fth 17912

This theorem is referenced by:  relfth  17916  isfth  17921  fthoppc  17930  fthsect  17932  fthinv  17933  fthmon  17934  fthepi  17935  ffthiso  17936  cofth  17942  inclfusubc  17948  fthcomf  49716  fthoppf  49723