Theorem fullpropd 17882

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 17870	. 2 ⊢ Rel (𝐴 Full 𝐶)
2		relfull 17870	. 2 ⊢ Rel (𝐵 Full 𝐷)
3		fullpropd.1	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
4	3	homfeqbas 17649	. . . . . . 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 2726	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		eqid 2726	. . . . . . . . 9 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
9		eqid 2726	. . . . . . . . 9 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
10		fullpropd.3	. . . . . . . . . 10 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
11	10	ad3antrrr 727	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → (Homf ‘𝐶) = (Homf ‘𝐷))
12		eqid 2726	. . . . . . . . . . 11 ⊢ (Base‘𝐴) = (Base‘𝐴)
13		simpllr 773	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓(𝐴 Func 𝐶)𝑔)
14	12, 7, 13	funcf1 17825	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑓:(Base‘𝐴)⟶(Base‘𝐶))
15		simplr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓(𝐴 Func 𝐶)𝑔) ∧ 𝑥 ∈ (Base‘𝐴)) ∧ 𝑦 ∈ (Base‘𝐴)) → 𝑥 ∈ (Base‘𝐴))
16	14, 15	ffvelcdmd 7081	. . . . . . . . 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 7081	. . . . . . . . 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 17650	. . . . . . . 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 2737	. . . . . . 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 3321	. . . . . 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 3321	. . . . 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 17862	. . . . . 6 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
31	30	breqd 5152	. . . . 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 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 17872	. . 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 2726	. . . 4 ⊢ (Base‘𝐵) = (Base‘𝐵)
36	35, 9	isfull 17872	. . 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 5787	1 ⊢ (𝜑 → (𝐴 Full 𝐶) = (𝐵 Full 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055   class class class wbr 5141  ran crn 5670  ‘cfv 6537  (class class class)co 7405  Basecbs 17153  Hom chom 17217  Hom_f chomf 17619  comp_fccomf 17620   Func cfunc 17813   Full cful 17864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824  df-ixp 8894  df-cat 17621  df-cid 17622  df-homf 17623  df-comf 17624  df-func 17817  df-full 17866

This theorem is referenced by: (None)