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Theorem fthpropd 17813
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 17801 . 2 Rel (𝐴 Faith 𝐶)
2 relfth 17801 . 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 17792 . . . . 5 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
1211breqd 5117 . . . 4 (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔𝑓(𝐵 Func 𝐷)𝑔))
133homfeqbas 17581 . . . . 5 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
1413raleqdv 3312 . . . . 5 (𝜑 → (∀𝑦 ∈ (Base‘𝐴)Fun (𝑥𝑔𝑦) ↔ ∀𝑦 ∈ (Base‘𝐵)Fun (𝑥𝑔𝑦)))
1513, 14raleqbidv 3318 . . . 4 (𝜑 → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun (𝑥𝑔𝑦) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun (𝑥𝑔𝑦)))
1612, 15anbi12d 632 . . 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 2733 . . . 4 (Base‘𝐴) = (Base‘𝐴)
1817isfth 17806 . . 3 (𝑓(𝐴 Faith 𝐶)𝑔 ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun (𝑥𝑔𝑦)))
19 eqid 2733 . . . 4 (Base‘𝐵) = (Base‘𝐵)
2019isfth 17806 . . 3 (𝑓(𝐵 Faith 𝐷)𝑔 ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun (𝑥𝑔𝑦)))
2116, 18, 203bitr4g 314 . 2 (𝜑 → (𝑓(𝐴 Faith 𝐶)𝑔𝑓(𝐵 Faith 𝐷)𝑔))
221, 2, 21eqbrrdiv 5751 1 (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061   class class class wbr 5106  ccnv 5633  Fun wfun 6491  cfv 6497  (class class class)co 7358  Basecbs 17088  Homf chomf 17551  compfccomf 17552   Func cfunc 17745   Faith cfth 17795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770  df-ixp 8839  df-cat 17553  df-cid 17554  df-homf 17555  df-comf 17556  df-func 17749  df-fth 17797
This theorem is referenced by: (None)
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