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Theorem fthpropd 17871
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 → (Homf𝐴) = (Homf𝐵))
fullpropd.2 (𝜑 → (compf𝐴) = (compf𝐵))
fullpropd.3 (𝜑 → (Homf𝐶) = (Homf𝐷))
fullpropd.4 (𝜑 → (compf𝐶) = (compf𝐷))
fullpropd.a (𝜑𝐴𝑉)
fullpropd.b (𝜑𝐵𝑉)
fullpropd.c (𝜑𝐶𝑉)
fullpropd.d (𝜑𝐷𝑉)
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.
StepHypRef Expression
1 relfth 17859 . 2 Rel (𝐴 Faith 𝐶)
2 relfth 17859 . 2 Rel (𝐵 Faith 𝐷)
3 fullpropd.1 . . . . . 6 (𝜑 → (Homf𝐴) = (Homf𝐵))
4 fullpropd.2 . . . . . 6 (𝜑 → (compf𝐴) = (compf𝐵))
5 fullpropd.3 . . . . . 6 (𝜑 → (Homf𝐶) = (Homf𝐷))
6 fullpropd.4 . . . . . 6 (𝜑 → (compf𝐶) = (compf𝐷))
7 fullpropd.a . . . . . 6 (𝜑𝐴𝑉)
8 fullpropd.b . . . . . 6 (𝜑𝐵𝑉)
9 fullpropd.c . . . . . 6 (𝜑𝐶𝑉)
10 fullpropd.d . . . . . 6 (𝜑𝐷𝑉)
113, 4, 5, 6, 7, 8, 9, 10funcpropd 17850 . . . . 5 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
1211breqd 5159 . . . 4 (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔𝑓(𝐵 Func 𝐷)𝑔))
133homfeqbas 17639 . . . . 5 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
1413raleqdv 3325 . . . . 5 (𝜑 → (∀𝑦 ∈ (Base‘𝐴)Fun (𝑥𝑔𝑦) ↔ ∀𝑦 ∈ (Base‘𝐵)Fun (𝑥𝑔𝑦)))
1513, 14raleqbidv 3342 . . . 4 (𝜑 → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun (𝑥𝑔𝑦) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun (𝑥𝑔𝑦)))
1612, 15anbi12d 631 . . 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 2732 . . . 4 (Base‘𝐴) = (Base‘𝐴)
1817isfth 17864 . . 3 (𝑓(𝐴 Faith 𝐶)𝑔 ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun (𝑥𝑔𝑦)))
19 eqid 2732 . . . 4 (Base‘𝐵) = (Base‘𝐵)
2019isfth 17864 . . 3 (𝑓(𝐵 Faith 𝐷)𝑔 ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun (𝑥𝑔𝑦)))
2116, 18, 203bitr4g 313 . 2 (𝜑 → (𝑓(𝐴 Faith 𝐶)𝑔𝑓(𝐵 Faith 𝐷)𝑔))
221, 2, 21eqbrrdiv 5794 1 (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3061   class class class wbr 5148  ccnv 5675  Fun wfun 6537  cfv 6543  (class class class)co 7408  Basecbs 17143  Homf chomf 17609  compfccomf 17610   Func cfunc 17803   Faith cfth 17853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-map 8821  df-ixp 8891  df-cat 17611  df-cid 17612  df-homf 17613  df-comf 17614  df-func 17807  df-fth 17855
This theorem is referenced by: (None)
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