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Theorem fullpropd 17967
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same full functors. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fullpropd.1 (𝜑 → (Homf𝐴) = (Homf𝐵))
fullpropd.2 (𝜑 → (compf𝐴) = (compf𝐵))
fullpropd.3 (𝜑 → (Homf𝐶) = (Homf𝐷))
fullpropd.4 (𝜑 → (compf𝐶) = (compf𝐷))
fullpropd.a (𝜑𝐴𝑉)
fullpropd.b (𝜑𝐵𝑉)
fullpropd.c (𝜑𝐶𝑉)
fullpropd.d (𝜑𝐷𝑉)
Assertion
Ref Expression
fullpropd (𝜑 → (𝐴 Full 𝐶) = (𝐵 Full 𝐷))

Proof of Theorem fullpropd
Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfull 17955 . 2 Rel (𝐴 Full 𝐶)
2 relfull 17955 . 2 Rel (𝐵 Full 𝐷)
3 fullpropd.1 . . . . . . . 8 (𝜑 → (Homf𝐴) = (Homf𝐵))
43homfeqbas 17739 . . . . . . 7 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
54adantr 480 . . . . . 6 ((𝜑𝑓(𝐴 Func 𝐶)𝑔) → (Base‘𝐴) = (Base‘𝐵))
65adantr 480 . . . . . . 7 (((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
7 eqid 2737 . . . . . . . . 9 (Base‘𝐶) = (Base‘𝐶)
8 eqid 2737 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
9 eqid 2737 . . . . . . . . 9 (Hom ‘𝐷) = (Hom ‘𝐷)
10 fullpropd.3 . . . . . . . . . 10 (𝜑 → (Homf𝐶) = (Homf𝐷))
1110ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (Homf𝐶) = (Homf𝐷))
12 eqid 2737 . . . . . . . . . . 11 (Base‘𝐴) = (Base‘𝐴)
13 simpllr 776 . . . . . . . . . . 11 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓(𝐴 Func 𝐶)𝑔)
1412, 7, 13funcf1 17911 . . . . . . . . . 10 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓:(Base‘𝐴)⟶(Base‘𝐶))
15 simplr 769 . . . . . . . . . 10 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑥 ∈ (Base‘𝐴))
1614, 15ffvelcdmd 7105 . . . . . . . . 9 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑓𝑥) ∈ (Base‘𝐶))
17 simpr 484 . . . . . . . . . 10 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑦 ∈ (Base‘𝐴))
1814, 17ffvelcdmd 7105 . . . . . . . . 9 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑓𝑦) ∈ (Base‘𝐶))
197, 8, 9, 11, 16, 18homfeqval 17740 . . . . . . . 8 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → ((𝑓𝑥)(Hom ‘𝐶)(𝑓𝑦)) = ((𝑓𝑥)(Hom ‘𝐷)(𝑓𝑦)))
2019eqeq2d 2748 . . . . . . 7 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐶)(𝑓𝑦)) ↔ ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐷)(𝑓𝑦))))
216, 20raleqbidva 3332 . . . . . 6 (((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) → (∀𝑦 ∈ (Base‘𝐴)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐶)(𝑓𝑦)) ↔ ∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐷)(𝑓𝑦))))
225, 21raleqbidva 3332 . . . . 5 ((𝜑𝑓(𝐴 Func 𝐶)𝑔) → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐶)(𝑓𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐷)(𝑓𝑦))))
2322pm5.32da 579 . . . 4 (𝜑 → ((𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐶)(𝑓𝑦))) ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐷)(𝑓𝑦)))))
24 fullpropd.2 . . . . . . 7 (𝜑 → (compf𝐴) = (compf𝐵))
25 fullpropd.4 . . . . . . 7 (𝜑 → (compf𝐶) = (compf𝐷))
26 fullpropd.a . . . . . . 7 (𝜑𝐴𝑉)
27 fullpropd.b . . . . . . 7 (𝜑𝐵𝑉)
28 fullpropd.c . . . . . . 7 (𝜑𝐶𝑉)
29 fullpropd.d . . . . . . 7 (𝜑𝐷𝑉)
303, 24, 10, 25, 26, 27, 28, 29funcpropd 17947 . . . . . 6 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
3130breqd 5154 . . . . 5 (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔𝑓(𝐵 Func 𝐷)𝑔))
3231anbi1d 631 . . . 4 (𝜑 → ((𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐷)(𝑓𝑦))) ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐷)(𝑓𝑦)))))
3323, 32bitrd 279 . . 3 (𝜑 → ((𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐶)(𝑓𝑦))) ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐷)(𝑓𝑦)))))
3412, 8isfull 17957 . . 3 (𝑓(𝐴 Full 𝐶)𝑔 ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐶)(𝑓𝑦))))
35 eqid 2737 . . . 4 (Base‘𝐵) = (Base‘𝐵)
3635, 9isfull 17957 . . 3 (𝑓(𝐵 Full 𝐷)𝑔 ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓𝑥)(Hom ‘𝐷)(𝑓𝑦))))
3733, 34, 363bitr4g 314 . 2 (𝜑 → (𝑓(𝐴 Full 𝐶)𝑔𝑓(𝐵 Full 𝐷)𝑔))
381, 2, 37eqbrrdiv 5804 1 (𝜑 → (𝐴 Full 𝐶) = (𝐵 Full 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061   class class class wbr 5143  ran crn 5686  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  Homf chomf 17709  compfccomf 17710   Func cfunc 17899   Full cful 17949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ixp 8938  df-cat 17711  df-cid 17712  df-homf 17713  df-comf 17714  df-func 17903  df-full 17951
This theorem is referenced by: (None)
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