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 → (Homf ‘𝐴) = (Homf ‘𝐵))
fullpropd.2	⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
fullpropd.3	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
fullpropd.4	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
fullpropd.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fullpropd.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
fullpropd.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
fullpropd.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)

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.

Step	Hyp	Ref	Expression
1		relfull 17955	. 2 ⊢ Rel (𝐴 Full 𝐶)
2		relfull 17955	. 2 ⊢ Rel (𝐵 Full 𝐷)
3		fullpropd.1	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
4	3	homfeqbas 17739	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) → (Base‘𝐴) = (Base‘𝐵))
6	5	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓(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 → (Homf ‘𝐶) = (Homf ‘𝐷))
11	10	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (Homf ‘𝐶) = (Homf ‘𝐷))
12		eqid 2737	. . . . . . . . . . 11 ⊢ (Base‘𝐴) = (Base‘𝐴)
13		simpllr 776	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓(𝐴 Func 𝐶)𝑔)
14	12, 7, 13	funcf1 17911	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓:(Base‘𝐴)⟶(Base‘𝐶))
15		simplr 769	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑥 ∈ (Base‘𝐴))
16	14, 15	ffvelcdmd 7105	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑓‘𝑥) ∈ (Base‘𝐶))
17		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑦 ∈ (Base‘𝐴))
18	14, 17	ffvelcdmd 7105	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑓‘𝑦) ∈ (Base‘𝐶))
19	7, 8, 9, 11, 16, 18	homfeqval 17740	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → ((𝑓‘𝑥)(Hom ‘𝐶)(𝑓‘𝑦)) = ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦)))
20	19	eqeq2d 2748	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐶)(𝑓‘𝑦)) ↔ ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦))))
21	6, 20	raleqbidva 3332	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) → (∀𝑦 ∈ (Base‘𝐴)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐶)(𝑓‘𝑦)) ↔ ∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦))))
22	5, 21	raleqbidva 3332	. . . . 5 ⊢ ((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐶)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦))))
23	22	pm5.32da 579	. . . 4 ⊢ (𝜑 → ((𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐶)(𝑓‘𝑦))) ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦)))))
24		fullpropd.2	. . . . . . 7 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
25		fullpropd.4	. . . . . . 7 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
26		fullpropd.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
27		fullpropd.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
28		fullpropd.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
29		fullpropd.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
30	3, 24, 10, 25, 26, 27, 28, 29	funcpropd 17947	. . . . . 6 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
31	30	breqd 5154	. . . . 5 ⊢ (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔 ↔ 𝑓(𝐵 Func 𝐷)𝑔))
32	31	anbi1d 631	. . . 4 ⊢ (𝜑 → ((𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦))) ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦)))))
33	23, 32	bitrd 279	. . 3 ⊢ (𝜑 → ((𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐶)(𝑓‘𝑦))) ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦)))))
34	12, 8	isfull 17957	. . 3 ⊢ (𝑓(𝐴 Full 𝐶)𝑔 ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐶)(𝑓‘𝑦))))
35		eqid 2737	. . . 4 ⊢ (Base‘𝐵) = (Base‘𝐵)
36	35, 9	isfull 17957	. . 3 ⊢ (𝑓(𝐵 Full 𝐷)𝑔 ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)ran (𝑥𝑔𝑦) = ((𝑓‘𝑥)(Hom ‘𝐷)(𝑓‘𝑦))))
37	33, 34, 36	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑓(𝐴 Full 𝐶)𝑔 ↔ 𝑓(𝐵 Full 𝐷)𝑔))
38	1, 2, 37	eqbrrdiv 5804	1 ⊢ (𝜑 → (𝐴 Full 𝐶) = (𝐵 Full 𝐷))

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  Hom_f chomf 17709  comp_fccomf 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)