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Theorem fthfunc 16614
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.
StepHypRef Expression
1 oveq1 6697 . . . 4 (𝑐 = 𝐶 → (𝑐 Faith 𝑑) = (𝐶 Faith 𝑑))
2 oveq1 6697 . . . 4 (𝑐 = 𝐶 → (𝑐 Func 𝑑) = (𝐶 Func 𝑑))
31, 2sseq12d 3667 . . 3 (𝑐 = 𝐶 → ((𝑐 Faith 𝑑) ⊆ (𝑐 Func 𝑑) ↔ (𝐶 Faith 𝑑) ⊆ (𝐶 Func 𝑑)))
4 oveq2 6698 . . . 4 (𝑑 = 𝐷 → (𝐶 Faith 𝑑) = (𝐶 Faith 𝐷))
5 oveq2 6698 . . . 4 (𝑑 = 𝐷 → (𝐶 Func 𝑑) = (𝐶 Func 𝐷))
64, 5sseq12d 3667 . . 3 (𝑑 = 𝐷 → ((𝐶 Faith 𝑑) ⊆ (𝐶 Func 𝑑) ↔ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)))
7 ovex 6718 . . . . . 6 (𝑐 Func 𝑑) ∈ V
8 simpl 472 . . . . . . . 8 ((𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦)) → 𝑓(𝑐 Func 𝑑)𝑔)
98ssopab2i 5032 . . . . . . 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 4747 . . . . . . 7 {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝑐 Func 𝑑)𝑔} ⊆ (𝑐 Func 𝑑)
119, 10sstri 3645 . . . . . 6 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))} ⊆ (𝑐 Func 𝑑)
127, 11ssexi 4836 . . . . 5 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))} ∈ V
13 df-fth 16612 . . . . . 6 Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))})
1413ovmpt4g 6825 . . . . 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))})
1512, 14mp3an3 1453 . . . 4 ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Faith 𝑑) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))})
1615, 11syl6eqss 3688 . . 3 ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Faith 𝑑) ⊆ (𝑐 Func 𝑑))
173, 6, 16vtocl2ga 3305 . 2 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷))
1813mpt2ndm0 6917 . . 3 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) = ∅)
19 0ss 4005 . . 3 ∅ ⊆ (𝐶 Func 𝐷)
2018, 19syl6eqss 3688 . 2 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷))
2117, 20pm2.61i 176 1 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  wss 3607  c0 3948   class class class wbr 4685  {copab 4745  ccnv 5142  Fun wfun 5920  cfv 5926  (class class class)co 6690  Basecbs 15904  Catccat 16372   Func cfunc 16561   Faith cfth 16610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-fth 16612
This theorem is referenced by:  relfth  16616  isfth  16621  fthoppc  16630  fthsect  16632  fthinv  16633  fthmon  16634  fthepi  16635  ffthiso  16636  cofth  16642  inclfusubc  42192
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