Theorem fullpropd 17190

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 17178	. 2 ⊢ Rel (𝐴 Full 𝐶)
2		relfull 17178	. 2 ⊢ Rel (𝐵 Full 𝐷)
3		fullpropd.1	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
4	3	homfeqbas 16966	. . . . . . 7 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
5	4	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) → (Base‘𝐴) = (Base‘𝐵))
6	5	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) → (Base‘𝐴) = (Base‘𝐵))
7		eqid 2821	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		eqid 2821	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
9		eqid 2821	. . . . . . . . 9 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10		fullpropd.3	. . . . . . . . . 10 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
11	10	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (Homf ‘𝐶) = (Homf ‘𝐷))
12		eqid 2821	. . . . . . . . . . 11 ⊢ (Base‘𝐴) = (Base‘𝐴)
13		simpllr 774	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓(𝐴 Func 𝐶)𝑔)
14	12, 7, 13	funcf1 17136	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓:(Base‘𝐴)⟶(Base‘𝐶))
15		simplr 767	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑥 ∈ (Base‘𝐴))
16	14, 15	ffvelrnd 6852	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (𝑓‘𝑥) ∈ (Base‘𝐶))
17		simpr 487	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑦 ∈ (Base‘𝐴))
18	14, 17	ffvelrnd 6852	. . . . . . . . 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 16967	. . . . . . . 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 2832	. . . . . . 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 3425	. . . . . 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 3425	. . . . 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 581	. . . 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 17170	. . . . . 6 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
31	30	breqd 5077	. . . . 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 281	. . 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 17180	. . 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 2821	. . . 4 ⊢ (Base‘𝐵) = (Base‘𝐵)
36	35, 9	isfull 17180	. . 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 316	. 2 ⊢ (𝜑 → (𝑓(𝐴 Full 𝐶)𝑔 ↔ 𝑓(𝐵 Full 𝐷)𝑔))
38	1, 2, 37	eqbrrdiv 5667	1 ⊢ (𝜑 → (𝐴 Full 𝐶) = (𝐵 Full 𝐷))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138   class class class wbr 5066  ran crn 5556  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  Hom chom 16576  Hom_f chomf 16937  comp_fccomf 16938   Func cfunc 17124   Full cful 17172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-map 8408  df-ixp 8462  df-cat 16939  df-cid 16940  df-homf 16941  df-comf 16942  df-func 17128  df-full 17174

This theorem is referenced by: (None)