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Theorem isfth 16621
Description: Value of the set of faithful functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypothesis
Ref Expression
isfth.b 𝐵 = (Base‘𝐶)
Assertion
Ref Expression
isfth (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦

Proof of Theorem isfth
Dummy variables 𝑐 𝑑 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fthfunc 16614 . . 3 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
21ssbri 4730 . 2 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
3 df-br 4686 . . . . . . 7 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4 funcrcl 16570 . . . . . . 7 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
53, 4sylbi 207 . . . . . 6 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6 oveq12 6699 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
76breqd 4696 . . . . . . . . 9 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝑓(𝑐 Func 𝑑)𝑔𝑓(𝐶 Func 𝐷)𝑔))
8 simpl 472 . . . . . . . . . . . 12 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑐 = 𝐶)
98fveq2d 6233 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10 isfth.b . . . . . . . . . . 11 𝐵 = (Base‘𝐶)
119, 10syl6eqr 2703 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = 𝐵)
1211raleqdv 3174 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦) ↔ ∀𝑦𝐵 Fun (𝑥𝑔𝑦)))
1311, 12raleqbidv 3182 . . . . . . . . 9 ((𝑐 = 𝐶𝑑 = 𝐷) → (∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦) ↔ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦)))
147, 13anbi12d 747 . . . . . . . 8 ((𝑐 = 𝐶𝑑 = 𝐷) → ((𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦)) ↔ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))))
1514opabbidv 4749 . . . . . . 7 ((𝑐 = 𝐶𝑑 = 𝐷) → {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))} = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))})
16 df-fth 16612 . . . . . . 7 Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))})
17 ovex 6718 . . . . . . . 8 (𝐶 Func 𝐷) ∈ V
18 simpl 472 . . . . . . . . . 10 ((𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦)) → 𝑓(𝐶 Func 𝐷)𝑔)
1918ssopab2i 5032 . . . . . . . . 9 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))} ⊆ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔}
20 opabss 4747 . . . . . . . . 9 {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔} ⊆ (𝐶 Func 𝐷)
2119, 20sstri 3645 . . . . . . . 8 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))} ⊆ (𝐶 Func 𝐷)
2217, 21ssexi 4836 . . . . . . 7 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))} ∈ V
2315, 16, 22ovmpt2a 6833 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Faith 𝐷) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))})
245, 23syl 17 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 Faith 𝐷) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))})
2524breqd 4696 . . . 4 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Faith 𝐷)𝐺𝐹{⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))}𝐺))
26 relfunc 16569 . . . . . 6 Rel (𝐶 Func 𝐷)
27 brrelex12 5189 . . . . . 6 ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
2826, 27mpan 706 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
29 breq12 4690 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓(𝐶 Func 𝐷)𝑔𝐹(𝐶 Func 𝐷)𝐺))
30 simpr 476 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
3130oveqd 6707 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
3231cnveqd 5330 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
3332funeqd 5948 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (Fun (𝑥𝑔𝑦) ↔ Fun (𝑥𝐺𝑦)))
34332ralbidv 3018 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦) ↔ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦)))
3529, 34anbi12d 747 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦)) ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦))))
36 eqid 2651 . . . . . 6 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))} = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))}
3735, 36brabga 5018 . . . . 5 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹{⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))}𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦))))
3828, 37syl 17 . . . 4 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹{⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))}𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦))))
3925, 38bitrd 268 . . 3 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦))))
4039bianabs 942 . 2 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦)))
412, 40biadan2 675 1 (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cop 4216   class class class wbr 4685  {copab 4745  ccnv 5142  Rel wrel 5148  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  ax-un 6991
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-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  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-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-func 16565  df-fth 16612
This theorem is referenced by:  isfth2  16622  fthpropd  16628  fthoppc  16630  fthres2b  16637  fthres2c  16638  fthres2  16639
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