Theorem fthpropd 17970

Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same faithful 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
fthpropd	⊢ (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷))

Proof of Theorem fthpropd

Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relfth 17958	. 2 ⊢ Rel (𝐴 Faith 𝐶)
2		relfth 17958	. 2 ⊢ Rel (𝐵 Faith 𝐷)
3		fullpropd.1	. . . . . 6 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))
4		fullpropd.2	. . . . . 6 ⊢ (𝜑 → (compf‘𝐴) = (compf‘𝐵))
5		fullpropd.3	. . . . . 6 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
6		fullpropd.4	. . . . . 6 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
7		fullpropd.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
8		fullpropd.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
9		fullpropd.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
10		fullpropd.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
11	3, 4, 5, 6, 7, 8, 9, 10	funcpropd 17949	. . . . 5 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
12	11	breqd 5116	. . . 4 ⊢ (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔 ↔ 𝑓(𝐵 Func 𝐷)𝑔))
13	3	homfeqbas 17742	. . . . 5 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
14	13	raleqdv 3323	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦)))
15	13, 14	raleqbidv 3339	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦)))
16	12, 15	anbi12d 643	. . 3 ⊢ (𝜑 → ((𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦)) ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦))))
17		eqid 2765	. . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴)
18	17	isfth 17963	. . 3 ⊢ (𝑓(𝐴 Faith 𝐶)𝑔 ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦)))
19		eqid 2765	. . . 4 ⊢ (Base‘𝐵) = (Base‘𝐵)
20	19	isfth 17963	. . 3 ⊢ (𝑓(𝐵 Faith 𝐷)𝑔 ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦)))
21	16, 18, 20	3bitr4g 317	. 2 ⊢ (𝜑 → (𝑓(𝐴 Faith 𝐶)𝑔 ↔ 𝑓(𝐵 Faith 𝐷)𝑔))
22	1, 2, 21	eqbrrdiv 5771	1 ⊢ (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079   class class class wbr 5105  ^◡ccnv 5651  Fun wfun 6519  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  Hom_f chomf 17712  comp_fccomf 17713   Func cfunc 17901   Faith cfth 17952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-ixp 8884  df-cat 17714  df-cid 17715  df-homf 17716  df-comf 17717  df-func 17905  df-fth 17954

This theorem is referenced by: (None)