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Theorem isfth 16397
 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 16390 . . 3 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
21ssbri 4627 . 2 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
3 df-br 4584 . . . . . . 7 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4 funcrcl 16346 . . . . . . 7 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
53, 4sylbi 206 . . . . . 6 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6 oveq12 6558 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
76breqd 4594 . . . . . . . . 9 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝑓(𝑐 Func 𝑑)𝑔𝑓(𝐶 Func 𝐷)𝑔))
8 simpl 472 . . . . . . . . . . . 12 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑐 = 𝐶)
98fveq2d 6107 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10 isfth.b . . . . . . . . . . 11 𝐵 = (Base‘𝐶)
119, 10syl6eqr 2662 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = 𝐵)
1211raleqdv 3121 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦) ↔ ∀𝑦𝐵 Fun (𝑥𝑔𝑦)))
1311, 12raleqbidv 3129 . . . . . . . . 9 ((𝑐 = 𝐶𝑑 = 𝐷) → (∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦) ↔ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦)))
147, 13anbi12d 743 . . . . . . . 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 4648 . . . . . . 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 16388 . . . . . . 7 Faith = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)Fun (𝑥𝑔𝑦))})
17 ovex 6577 . . . . . . . 8 (𝐶 Func 𝐷) ∈ V
18 simpl 472 . . . . . . . . . 10 ((𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦)) → 𝑓(𝐶 Func 𝐷)𝑔)
1918ssopab2i 4928 . . . . . . . . 9 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))} ⊆ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔}
20 opabss 4646 . . . . . . . . 9 {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔} ⊆ (𝐶 Func 𝐷)
2119, 20sstri 3577 . . . . . . . 8 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))} ⊆ (𝐶 Func 𝐷)
2217, 21ssexi 4731 . . . . . . 7 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))} ∈ V
2315, 16, 22ovmpt2a 6689 . . . . . 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 4594 . . . 4 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Faith 𝐷)𝐺𝐹{⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))}𝐺))
26 relfunc 16345 . . . . . 6 Rel (𝐶 Func 𝐷)
27 brrelex12 5079 . . . . . 6 ((Rel (𝐶 Func 𝐷) ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹 ∈ V ∧ 𝐺 ∈ V))
2826, 27mpan 702 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
29 breq12 4588 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓(𝐶 Func 𝐷)𝑔𝐹(𝐶 Func 𝐷)𝐺))
30 simpr 476 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
3130oveqd 6566 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
3231cnveqd 5220 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
3332funeqd 5825 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (Fun (𝑥𝑔𝑦) ↔ Fun (𝑥𝐺𝑦)))
34332ralbidv 2972 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦) ↔ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦)))
3529, 34anbi12d 743 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦)) ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦))))
36 eqid 2610 . . . . . 6 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))} = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝑔𝑦))}
3735, 36brabga 4914 . . . . 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 267 . . 3 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦))))
4039bianabs 920 . 2 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Faith 𝐷)𝐺 ↔ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦)))
412, 40biadan2 672 1 (𝐹(𝐶 Faith 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 Fun (𝑥𝐺𝑦)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⟨cop 4131   class class class wbr 4583  {copab 4642  ◡ccnv 5037  Rel wrel 5043  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Catccat 16148   Func cfunc 16337   Faith cfth 16386 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-func 16341  df-fth 16388 This theorem is referenced by:  isfth2  16398  fthpropd  16404  fthoppc  16406  fthres2b  16413  fthres2c  16414  fthres2  16415
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