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Theorem fullpropd 16801
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 16789 . 2 Rel (𝐴 Full 𝐶)
2 relfull 16789 . 2 Rel (𝐵 Full 𝐷)
3 fullpropd.1 . . . . . . . 8 (𝜑 → (Homf𝐴) = (Homf𝐵))
43homfeqbas 16577 . . . . . . 7 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
54adantr 472 . . . . . 6 ((𝜑𝑓(𝐴 Func 𝐶)𝑔) → (Base‘𝐴) = (Base‘𝐵))
65adantr 472 . . . . . . 7 (((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
7 eqid 2760 . . . . . . . . 9 (Base‘𝐶) = (Base‘𝐶)
8 eqid 2760 . . . . . . . . 9 (Hom ‘𝐶) = (Hom ‘𝐶)
9 eqid 2760 . . . . . . . . 9 (Hom ‘𝐷) = (Hom ‘𝐷)
10 fullpropd.3 . . . . . . . . . 10 (𝜑 → (Homf𝐶) = (Homf𝐷))
1110ad3antrrr 768 . . . . . . . . 9 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (Homf𝐶) = (Homf𝐷))
12 eqid 2760 . . . . . . . . . . 11 (Base‘𝐴) = (Base‘𝐴)
13 simpllr 817 . . . . . . . . . . 11 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓(𝐴 Func 𝐶)𝑔)
1412, 7, 13funcf1 16747 . . . . . . . . . 10 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓:(Base‘𝐴)⟶(Base‘𝐶))
15 simplr 809 . . . . . . . . . 10 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑥 ∈ (Base‘𝐴))
1614, 15ffvelrnd 6524 . . . . . . . . 9 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑓𝑥) ∈ (Base‘𝐶))
17 simpr 479 . . . . . . . . . 10 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑦 ∈ (Base‘𝐴))
1814, 17ffvelrnd 6524 . . . . . . . . 9 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑓𝑦) ∈ (Base‘𝐶))
197, 8, 9, 11, 16, 18homfeqval 16578 . . . . . . . 8 ((((𝜑𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → ((𝑓𝑥)(Hom ‘𝐶)(𝑓𝑦)) = ((𝑓𝑥)(Hom ‘𝐷)(𝑓𝑦)))
2019eqeq2d 2770 . . . . . . 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 3293 . . . . . 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 3293 . . . . 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 676 . . . 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 16781 . . . . . 6 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
3130breqd 4815 . . . . 5 (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔𝑓(𝐵 Func 𝐷)𝑔))
3231anbi1d 743 . . . 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 268 . . 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 16791 . . 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 2760 . . . 4 (Base‘𝐵) = (Base‘𝐵)
3635, 9isfull 16791 . . 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 303 . 2 (𝜑 → (𝑓(𝐴 Full 𝐶)𝑔𝑓(𝐵 Full 𝐷)𝑔))
381, 2, 37eqbrrdiv 5375 1 (𝜑 → (𝐴 Full 𝐶) = (𝐵 Full 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050   class class class wbr 4804  ran crn 5267  cfv 6049  (class class class)co 6814  Basecbs 16079  Hom chom 16174  Homf chomf 16548  compfccomf 16549   Func cfunc 16735   Full cful 16783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027  df-ixp 8077  df-cat 16550  df-cid 16551  df-homf 16552  df-comf 16553  df-func 16739  df-full 16785
This theorem is referenced by: (None)
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