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Theorem fullpropd 16352
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 16340 . 2 Rel (𝐴 Full 𝐶)
2 relfull 16340 . 2 Rel (𝐵 Full 𝐷)
3 fullpropd.1 . . . . . . . 8 (𝜑 → (Homf𝐴) = (Homf𝐵))
43homfeqbas 16128 . . . . . . 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 2610 . . . . . . . . 9 (Base‘𝐶) = (Base‘𝐶)
8 eqid 2610 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
9 eqid 2610 . . . . . . . . 9 (Hom ‘𝐷) = (Hom ‘𝐷)
10 fullpropd.3 . . . . . . . . . 10 (𝜑 → (Homf𝐶) = (Homf𝐷))
1110ad3antrrr 762 . . . . . . . . 9 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (Homf𝐶) = (Homf𝐷))
12 eqid 2610 . . . . . . . . . . 11 (Base‘𝐴) = (Base‘𝐴)
13 simpllr 795 . . . . . . . . . . 11 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓(𝐴 Func 𝐶)𝑔)
1412, 7, 13funcf1 16298 . . . . . . . . . 10 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓:(Base‘𝐴)⟶(Base‘𝐶))
15 simplr 788 . . . . . . . . . 10 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑥 ∈ (Base‘𝐴))
1614, 15ffvelrnd 6253 . . . . . . . . 9 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑓𝑥) ∈ (Base‘𝐶))
17 simpr 476 . . . . . . . . . 10 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑦 ∈ (Base‘𝐴))
1814, 17ffvelrnd 6253 . . . . . . . . 9 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑓𝑦) ∈ (Base‘𝐶))
197, 8, 9, 11, 16, 18homfeqval 16129 . . . . . . . 8 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → ((𝑓𝑥)(Hom ‘𝐶)(𝑓𝑦)) = ((𝑓𝑥)(Hom ‘𝐷)(𝑓𝑦)))
2019eqeq2d 2620 . . . . . . 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 3131 . . . . . 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 3131 . . . . 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 671 . . . 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 16332 . . . . . 6 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
3130breqd 4589 . . . . 5 (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔𝑓(𝐵 Func 𝐷)𝑔))
3231anbi1d 737 . . . 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 267 . . 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 16342 . . 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 2610 . . . 4 (Base‘𝐵) = (Base‘𝐵)
3635, 9isfull 16342 . . 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 302 . 2 (𝜑 → (𝑓(𝐴 Full 𝐶)𝑔𝑓(𝐵 Full 𝐷)𝑔))
381, 2, 37eqbrrdiv 5130 1 (𝜑 → (𝐴 Full 𝐶) = (𝐵 Full 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896   class class class wbr 4578  ran crn 5029  cfv 5790  (class class class)co 6527  Basecbs 15644  Hom chom 15728  Homf chomf 16099  compfccomf 16100   Func cfunc 16286   Full cful 16334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7037  df-2nd 7038  df-map 7724  df-ixp 7773  df-cat 16101  df-cid 16102  df-homf 16103  df-comf 16104  df-func 16290  df-full 16336
This theorem is referenced by: (None)
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