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Theorem isfull 17870
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 17866 . . 3 (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
21ssbri 5186 . 2 (𝐹(𝐶 Full 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
3 df-br 5142 . . . . . . 7 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
4 funcrcl 17820 . . . . . . 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 7413 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝑐 Func 𝑑) = (𝐶 Func 𝐷))
76breqd 5152 . . . . . . . . 9 ((𝑐 = 𝐶𝑑 = 𝐷) → (𝑓(𝑐 Func 𝑑)𝑔𝑓(𝐶 Func 𝐷)𝑔))
8 simpl 482 . . . . . . . . . . . 12 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑐 = 𝐶)
98fveq2d 6888 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = (Base‘𝐶))
10 isfull.b . . . . . . . . . . 11 𝐵 = (Base‘𝐶)
119, 10eqtr4di 2784 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (Base‘𝑐) = 𝐵)
12 simpr 484 . . . . . . . . . . . . . . 15 ((𝑐 = 𝐶𝑑 = 𝐷) → 𝑑 = 𝐷)
1312fveq2d 6888 . . . . . . . . . . . . . 14 ((𝑐 = 𝐶𝑑 = 𝐷) → (Hom ‘𝑑) = (Hom ‘𝐷))
14 isfull.j . . . . . . . . . . . . . 14 𝐽 = (Hom ‘𝐷)
1513, 14eqtr4di 2784 . . . . . . . . . . . . 13 ((𝑐 = 𝐶𝑑 = 𝐷) → (Hom ‘𝑑) = 𝐽)
1615oveqd 7421 . . . . . . . . . . . 12 ((𝑐 = 𝐶𝑑 = 𝐷) → ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)))
1716eqeq2d 2737 . . . . . . . . . . 11 ((𝑐 = 𝐶𝑑 = 𝐷) → (ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)) ↔ ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦))))
1811, 17raleqbidv 3336 . . . . . . . . . 10 ((𝑐 = 𝐶𝑑 = 𝐷) → (∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)) ↔ ∀𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦))))
1911, 18raleqbidv 3336 . . . . . . . . 9 ((𝑐 = 𝐶𝑑 = 𝐷) → (∀𝑥 ∈ (Base‘𝑐)∀𝑦 ∈ (Base‘𝑐)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝑑)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦))))
207, 19anbi12d 630 . . . . . . . 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 5207 . . . . . . 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 17864 . . . . . . 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 7437 . . . . . . . 8 (𝐶 Func 𝐷) ∈ V
24 simpl 482 . . . . . . . . . 10 ((𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦))) → 𝑓(𝐶 Func 𝐷)𝑔)
2524ssopab2i 5543 . . . . . . . . 9 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))} ⊆ {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔}
26 opabss 5205 . . . . . . . . 9 {⟨𝑓, 𝑔⟩ ∣ 𝑓(𝐶 Func 𝐷)𝑔} ⊆ (𝐶 Func 𝐷)
2725, 26sstri 3986 . . . . . . . 8 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))} ⊆ (𝐶 Func 𝐷)
2823, 27ssexi 5315 . . . . . . 7 {⟨𝑓, 𝑔⟩ ∣ (𝑓(𝐶 Func 𝐷)𝑔 ∧ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝑔𝑦) = ((𝑓𝑥)𝐽(𝑓𝑦)))} ∈ V
2921, 22, 28ovmpoa 7558 . . . . . 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 5152 . . . 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 17819 . . . . . 6 Rel (𝐶 Func 𝐷)
3332brrelex12i 5724 . . . . 5 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
34 breq12 5146 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓(𝐶 Func 𝐷)𝑔𝐹(𝐶 Func 𝐷)𝐺))
35 simpr 484 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑔 = 𝐺)
3635oveqd 7421 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑥𝑔𝑦) = (𝑥𝐺𝑦))
3736rneqd 5930 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → ran (𝑥𝑔𝑦) = ran (𝑥𝐺𝑦))
38 simpl 482 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
3938fveq1d 6886 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑥) = (𝐹𝑥))
4038fveq1d 6886 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓𝑦) = (𝐹𝑦))
4139, 40oveq12d 7422 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝐽(𝑓𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)))
4237, 41eqeq12d 2742 . . . . . . . 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 630 . . . . . 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 2726 . . . . . 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 5527 . . . . 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 541 . 2 (𝐹(𝐶 Func 𝐷)𝐺 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥𝐵𝑦𝐵 ran (𝑥𝐺𝑦) = ((𝐹𝑥)𝐽(𝐹𝑦))))
502, 49biadanii 819 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 395   = wceq 1533  wcel 2098  wral 3055  Vcvv 3468  cop 4629   class class class wbr 5141  {copab 5203  ran crn 5670  cfv 6536  (class class class)co 7404  Basecbs 17151  Hom chom 17215  Catccat 17615   Func cfunc 17811   Full cful 17862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-func 17815  df-full 17864
This theorem is referenced by:  isfull2  17871  fullpropd  17880  fulloppc  17882  fullres2c  17899
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