Theorem fthpropd 17883

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 17871	. 2 ⊢ Rel (𝐴 Faith 𝐶)
2		relfth 17871	. 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 17862	. . . . 5 ⊢ (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
12	11	breqd 5152	. . . 4 ⊢ (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔 ↔ 𝑓(𝐵 Func 𝐷)𝑔))
13	3	homfeqbas 17649	. . . . 5 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
14	13	raleqdv 3319	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦)))
15	13, 14	raleqbidv 3336	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦)))
16	12, 15	anbi12d 630	. . 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 2726	. . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴)
18	17	isfth 17876	. . 3 ⊢ (𝑓(𝐴 Faith 𝐶)𝑔 ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun ◡(𝑥𝑔𝑦)))
19		eqid 2726	. . . 4 ⊢ (Base‘𝐵) = (Base‘𝐵)
20	19	isfth 17876	. . 3 ⊢ (𝑓(𝐵 Faith 𝐷)𝑔 ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun ◡(𝑥𝑔𝑦)))
21	16, 18, 20	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑓(𝐴 Faith 𝐶)𝑔 ↔ 𝑓(𝐵 Faith 𝐷)𝑔))
22	1, 2, 21	eqbrrdiv 5787	1 ⊢ (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055   class class class wbr 5141  ^◡ccnv 5668  Fun wfun 6531  ‘cfv 6537  (class class class)co 7405  Basecbs 17153  Hom_f chomf 17619  comp_fccomf 17620   Func cfunc 17813   Faith cfth 17865

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-fth 17867

This theorem is referenced by: (None)