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Theorem fthfunc 17006
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 7023 . . . 4 (𝑐 = 𝐶 → (𝑐 Faith 𝑑) = (𝐶 Faith 𝑑))
2 oveq1 7023 . . . 4 (𝑐 = 𝐶 → (𝑐 Func 𝑑) = (𝐶 Func 𝑑))
31, 2sseq12d 3921 . . 3 (𝑐 = 𝐶 → ((𝑐 Faith 𝑑) ⊆ (𝑐 Func 𝑑) ↔ (𝐶 Faith 𝑑) ⊆ (𝐶 Func 𝑑)))
4 oveq2 7024 . . . 4 (𝑑 = 𝐷 → (𝐶 Faith 𝑑) = (𝐶 Faith 𝐷))
5 oveq2 7024 . . . 4 (𝑑 = 𝐷 → (𝐶 Func 𝑑) = (𝐶 Func 𝐷))
64, 5sseq12d 3921 . . 3 (𝑑 = 𝐷 → ((𝐶 Faith 𝑑) ⊆ (𝐶 Func 𝑑) ↔ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)))
7 ovex 7048 . . . . . 6 (𝑐 Func 𝑑) ∈ V
8 simpl 483 . . . . . . . 8 ((𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦)) → 𝑓(𝑐 Func 𝑑)𝑔)
98ssopab2i 5325 . . . . . . 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 5026 . . . . . . 7 {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝑐 Func 𝑑)𝑔} ⊆ (𝑐 Func 𝑑)
119, 10sstri 3898 . . . . . 6 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))} ⊆ (𝑐 Func 𝑑)
127, 11ssexi 5117 . . . . 5 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))} ∈ V
13 df-fth 17004 . . . . . 6 Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))})
1413ovmpt4g 7153 . . . . 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 1442 . . . 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 3942 . . 3 ((𝑐 ∈ Cat ∧ 𝑑 ∈ Cat) → (𝑐 Faith 𝑑) ⊆ (𝑐 Func 𝑑))
173, 6, 16vtocl2ga 3518 . 2 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷))
1813mpondm0 7245 . . 3 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) = ∅)
19 0ss 4270 . . 3 ∅ ⊆ (𝐶 Func 𝐷)
2018, 19syl6eqss 3942 . 2 (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷))
2117, 20pm2.61i 183 1 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1522  wcel 2081  wral 3105  Vcvv 3437  wss 3859  c0 4211   class class class wbr 4962  {copab 5024  ccnv 5442  Fun wfun 6219  cfv 6225  (class class class)co 7016  Basecbs 16312  Catccat 16764   Func cfunc 16953   Faith cfth 17002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-iota 6189  df-fun 6227  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-fth 17004
This theorem is referenced by:  relfth  17008  isfth  17013  fthoppc  17022  fthsect  17024  fthinv  17025  fthmon  17026  fthepi  17027  ffthiso  17028  cofth  17034  inclfusubc  43636
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