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Theorem fullpropd 17974
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 17962 . 2 Rel (𝐴 Full 𝐶)
2 relfull 17962 . 2 Rel (𝐵 Full 𝐷)
3 fullpropd.1 . . . . . . . 8 (𝜑 → (Homf𝐴) = (Homf𝐵))
43homfeqbas 17741 . . . . . . 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 2735 . . . . . . . . 9 (Base‘𝐶) = (Base‘𝐶)
8 eqid 2735 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
9 eqid 2735 . . . . . . . . 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 2735 . . . . . . . . . . 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 17917 . . . . . . . . . 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 17742 . . . . . . . 8 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → ((𝑓𝑥)(Hom ‘𝐶)(𝑓𝑦)) = ((𝑓𝑥)(Hom ‘𝐷)(𝑓𝑦)))
2019eqeq2d 2746 . . . . . . 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 3330 . . . . . 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 3330 . . . . 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 17954 . . . . . 6 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
3130breqd 5159 . . . . 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 17964 . . 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 2735 . . . 4 (Base‘𝐵) = (Base‘𝐵)
3635, 9isfull 17964 . . 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 5807 1 (𝜑 → (𝐴 Full 𝐶) = (𝐵 Full 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059   class class class wbr 5148  ran crn 5690  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  Homf chomf 17711  compfccomf 17712   Func cfunc 17905   Full cful 17956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-ixp 8937  df-cat 17713  df-cid 17714  df-homf 17715  df-comf 17716  df-func 17909  df-full 17958
This theorem is referenced by: (None)
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