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Theorem fthpropd 16785
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 16773 . 2 Rel (𝐴 Faith 𝐶)
2 relfth 16773 . 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 16764 . . . . 5 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
1211breqd 4855 . . . 4 (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔𝑓(𝐵 Func 𝐷)𝑔))
133homfeqbas 16560 . . . . 5 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
1413raleqdv 3333 . . . . 5 (𝜑 → (∀𝑦 ∈ (Base‘𝐴)Fun (𝑥𝑔𝑦) ↔ ∀𝑦 ∈ (Base‘𝐵)Fun (𝑥𝑔𝑦)))
1513, 14raleqbidv 3341 . . . 4 (𝜑 → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun (𝑥𝑔𝑦) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun (𝑥𝑔𝑦)))
1612, 15anbi12d 618 . . 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 2806 . . . 4 (Base‘𝐴) = (Base‘𝐴)
1817isfth 16778 . . 3 (𝑓(𝐴 Faith 𝐶)𝑔 ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun (𝑥𝑔𝑦)))
19 eqid 2806 . . . 4 (Base‘𝐵) = (Base‘𝐵)
2019isfth 16778 . . 3 (𝑓(𝐵 Faith 𝐷)𝑔 ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun (𝑥𝑔𝑦)))
2116, 18, 203bitr4g 305 . 2 (𝜑 → (𝑓(𝐴 Faith 𝐶)𝑔𝑓(𝐵 Faith 𝐷)𝑔))
221, 2, 21eqbrrdiv 5420 1 (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  wral 3096   class class class wbr 4844  ccnv 5310  Fun wfun 6095  cfv 6101  (class class class)co 6874  Basecbs 16068  Homf chomf 16531  compfccomf 16532   Func cfunc 16718   Faith cfth 16767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-1st 7398  df-2nd 7399  df-map 8094  df-ixp 8146  df-cat 16533  df-cid 16534  df-homf 16535  df-comf 16536  df-func 16722  df-fth 16769
This theorem is referenced by: (None)
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