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Theorem isfull 17797
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 17793 . . 3 (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
21ssbri 5150 . 2 (𝐹(𝐶 Full 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
3 df-br 5106 . . . . . . 7 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4 funcrcl 17749 . . . . . . 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 7366 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
76breqd 5116 . . . . . . . . 9 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝑓(𝑐 Func 𝑑)𝑔𝑓(𝐶 Func 𝐷)𝑔))
8 simpl 483 . . . . . . . . . . . 12 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑐 = 𝐶)
98fveq2d 6846 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10 isfull.b . . . . . . . . . . 11 𝐵 = (Base‘𝐶)
119, 10eqtr4di 2794 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = 𝐵)
12 simpr 485 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑑 = 𝐷)
1312fveq2d 6846 . . . . . . . . . . . . . 14 ((𝑐 = 𝐶𝑑 = 𝐷) → (Hom ‘𝑑) = (Hom ‘𝐷))
14 isfull.j . . . . . . . . . . . . . 14 𝐽 = (Hom ‘𝐷)
1513, 14eqtr4di 2794 . . . . . . . . . . . . 13 ((𝑐 = 𝐶𝑑 = 𝐷) → (Hom ‘𝑑) = 𝐽)
1615oveqd 7374 . . . . . . . . . . . 12 ((𝑐 = 𝐶𝑑 = 𝐷) → ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)))
1716eqeq2d 2747 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑑 = 𝐷) → (ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)) ↔ ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦))))
1811, 17raleqbidv 3319 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)) ↔ ∀𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦))))
1911, 18raleqbidv 3319 . . . . . . . . 9 ((𝑐 = 𝐶𝑑 = 𝐷) → (∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦))))
207, 19anbi12d 631 . . . . . . . 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 5171 . . . . . . 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 17791 . . . . . . 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 7390 . . . . . . . 8 (𝐶 Func 𝐷) ∈ V
24 simpl 483 . . . . . . . . . 10 ((𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦))) → 𝑓(𝐶 Func 𝐷)𝑔)
2524ssopab2i 5507 . . . . . . . . 9 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))} ⊆ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔}
26 opabss 5169 . . . . . . . . 9 {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔} ⊆ (𝐶 Func 𝐷)
2725, 26sstri 3953 . . . . . . . 8 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))} ⊆ (𝐶 Func 𝐷)
2823, 27ssexi 5279 . . . . . . 7 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))} ∈ V
2921, 22, 28ovmpoa 7510 . . . . . 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 5116 . . . 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 17748 . . . . . 6 Rel (𝐶 Func 𝐷)
3332brrelex12i 5687 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
34 breq12 5110 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓(𝐶 Func 𝐷)𝑔𝐹(𝐶 Func 𝐷)𝐺))
35 simpr 485 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
3635oveqd 7374 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
3736rneqd 5893 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → ran (𝑥𝑔𝑦) = ran (𝑥𝐺𝑦))
38 simpl 483 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
3938fveq1d 6844 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑥) = (𝐹𝑥))
4038fveq1d 6844 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑦) = (𝐹𝑦))
4139, 40oveq12d 7375 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝐽(𝑓𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
4237, 41eqeq12d 2752 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → (ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)) ↔ ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
43422ralbidv 3212 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
4434, 43anbi12d 631 . . . . . 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 2736 . . . . . 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 5491 . . . . 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 278 . . 3 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦)))))
4948bianabs 542 . 2 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
502, 49biadanii 820 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 396   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cop 4592   class class class wbr 5105  {copab 5167  ran crn 5634  cfv 6496  (class class class)co 7357  Basecbs 17083  Hom chom 17144  Catccat 17544   Func cfunc 17740   Full cful 17789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-func 17744  df-full 17791
This theorem is referenced by:  isfull2  17798  fullpropd  17807  fulloppc  17809  fullres2c  17826
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