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Theorem isfull 17802
Description: Value of the set of full functors between two categories. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
isfull.b 𝐵 = (Base‘𝐶)
isfull.j 𝐽 = (Hom ‘𝐷)
Assertion
Ref Expression
isfull (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐽,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦

Proof of Theorem isfull
Dummy variables 𝑐 𝑑 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fullfunc 17798 . . 3 (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
21ssbri 5151 . 2 (𝐹(𝐶 Full 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
3 df-br 5107 . . . . . . 7 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4 funcrcl 17754 . . . . . . 7 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
53, 4sylbi 216 . . . . . 6 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
6 oveq12 7367 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
76breqd 5117 . . . . . . . . 9 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝑓(𝑐 Func 𝑑)𝑔𝑓(𝐶 Func 𝐷)𝑔))
8 simpl 484 . . . . . . . . . . . 12 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑐 = 𝐶)
98fveq2d 6847 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10 isfull.b . . . . . . . . . . 11 𝐵 = (Base‘𝐶)
119, 10eqtr4di 2791 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = 𝐵)
12 simpr 486 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑑 = 𝐷)
1312fveq2d 6847 . . . . . . . . . . . . . 14 ((𝑐 = 𝐶𝑑 = 𝐷) → (Hom ‘𝑑) = (Hom ‘𝐷))
14 isfull.j . . . . . . . . . . . . . 14 𝐽 = (Hom ‘𝐷)
1513, 14eqtr4di 2791 . . . . . . . . . . . . 13 ((𝑐 = 𝐶𝑑 = 𝐷) → (Hom ‘𝑑) = 𝐽)
1615oveqd 7375 . . . . . . . . . . . 12 ((𝑐 = 𝐶𝑑 = 𝐷) → ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)))
1716eqeq2d 2744 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑑 = 𝐷) → (ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)) ↔ ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦))))
1811, 17raleqbidv 3318 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)) ↔ ∀𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦))))
1911, 18raleqbidv 3318 . . . . . . . . 9 ((𝑐 = 𝐶𝑑 = 𝐷) → (∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦))))
207, 19anbi12d 632 . . . . . . . 8 ((𝑐 = 𝐶𝑑 = 𝐷) → ((𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦))) ↔ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))))
2120opabbidv 5172 . . . . . . 7 ((𝑐 = 𝐶𝑑 = 𝐷) → {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))})
22 df-full 17796 . . . . . . 7 Full = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝑐 Func 𝑑)𝑔 ∧ ∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)))})
23 ovex 7391 . . . . . . . 8 (𝐶 Func 𝐷) ∈ V
24 simpl 484 . . . . . . . . . 10 ((𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦))) → 𝑓(𝐶 Func 𝐷)𝑔)
2524ssopab2i 5508 . . . . . . . . 9 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))} ⊆ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔}
26 opabss 5170 . . . . . . . . 9 {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔} ⊆ (𝐶 Func 𝐷)
2725, 26sstri 3954 . . . . . . . 8 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))} ⊆ (𝐶 Func 𝐷)
2823, 27ssexi 5280 . . . . . . 7 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))} ∈ V
2921, 22, 28ovmpoa 7511 . . . . . 6 ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Full 𝐷) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))})
305, 29syl 17 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐶 Full 𝐷) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))})
3130breqd 5117 . . . 4 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Full 𝐷)𝐺𝐹{⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))}𝐺))
32 relfunc 17753 . . . . . 6 Rel (𝐶 Func 𝐷)
3332brrelex12i 5688 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
34 breq12 5111 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓(𝐶 Func 𝐷)𝑔𝐹(𝐶 Func 𝐷)𝐺))
35 simpr 486 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
3635oveqd 7375 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
3736rneqd 5894 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → ran (𝑥𝑔𝑦) = ran (𝑥𝐺𝑦))
38 simpl 484 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
3938fveq1d 6845 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑥) = (𝐹𝑥))
4038fveq1d 6845 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑦) = (𝐹𝑦))
4139, 40oveq12d 7376 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝐽(𝑓𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
4237, 41eqeq12d 2749 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)) ↔ ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
43422ralbidv 3209 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
4434, 43anbi12d 632 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦))) ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦)))))
45 eqid 2733 . . . . . 6 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))} = {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))}
4644, 45brabga 5492 . . . . 5 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹{⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))}𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦)))))
4733, 46syl 17 . . . 4 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹{⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))}𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦)))))
4831, 47bitrd 279 . . 3 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦)))))
4948bianabs 543 . 2 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
502, 49biadanii 821 1 (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  Vcvv 3444  cop 4593   class class class wbr 5106  {copab 5168  ran crn 5635  cfv 6497  (class class class)co 7358  Basecbs 17088  Hom chom 17149  Catccat 17549   Func cfunc 17745   Full cful 17794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-func 17749  df-full 17796
This theorem is referenced by:  isfull2  17803  fullpropd  17812  fulloppc  17814  fullres2c  17831
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