Theorem fullpropd 17552

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 17540	. 2 ⊢ Rel (𝐴 Full 𝐶)
2		relfull 17540	. 2 ⊢ Rel (𝐵 Full 𝐷)
3		fullpropd.1	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
4	3	homfeqbas 17322	. . . . . . 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 2738	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		eqid 2738	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
9		eqid 2738	. . . . . . . . 9 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10		fullpropd.3	. . . . . . . . . 10 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
11	10	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (Homf ‘𝐶) = (Homf ‘𝐷))
12		eqid 2738	. . . . . . . . . . 11 ⊢ (Base‘𝐴) = (Base‘𝐴)
13		simpllr 772	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓(𝐴 Func 𝐶)𝑔)
14	12, 7, 13	funcf1 17497	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓:(Base‘𝐴)⟶(Base‘𝐶))
15		simplr 765	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑥 ∈ (Base‘𝐴))
16	14, 15	ffvelrnd 6944	. . . . . . . . 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	ffvelrnd 6944	. . . . . . . . 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 17323	. . . . . . . 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 2749	. . . . . . 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 3345	. . . . . 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 3345	. . . . 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 578	. . . 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 17532	. . . . . 6 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
31	30	breqd 5081	. . . . 5 ⊢ (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔 ↔ 𝑓(𝐵 Func 𝐷)𝑔))
32	31	anbi1d 629	. . . 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 278	. . 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 17542	. . 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 2738	. . . 4 ⊢ (Base‘𝐵) = (Base‘𝐵)
36	35, 9	isfull 17542	. . 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 313	. 2 ⊢ (𝜑 → (𝑓(𝐴 Full 𝐶)𝑔 ↔ 𝑓(𝐵 Full 𝐷)𝑔))
38	1, 2, 37	eqbrrdiv 5693	1 ⊢ (𝜑 → (𝐴 Full 𝐶) = (𝐵 Full 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063   class class class wbr 5070  ran crn 5581  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  Hom chom 16899  Hom_f chomf 17292  comp_fccomf 17293   Func cfunc 17485   Full cful 17534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575  df-ixp 8644  df-cat 17294  df-cid 17295  df-homf 17296  df-comf 17297  df-func 17489  df-full 17536

This theorem is referenced by: (None)