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Theorem fthpropd 17186
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 17174 . 2 Rel (𝐴 Faith 𝐶)
2 relfth 17174 . 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 17165 . . . . 5 (𝜑 → (𝐴 Func 𝐶) = (𝐵 Func 𝐷))
1211breqd 5044 . . . 4 (𝜑 → (𝑓(𝐴 Func 𝐶)𝑔𝑓(𝐵 Func 𝐷)𝑔))
133homfeqbas 16961 . . . . 5 (𝜑 → (Base‘𝐴) = (Base‘𝐵))
1413raleqdv 3367 . . . . 5 (𝜑 → (∀𝑦 ∈ (Base‘𝐴)Fun (𝑥𝑔𝑦) ↔ ∀𝑦 ∈ (Base‘𝐵)Fun (𝑥𝑔𝑦)))
1513, 14raleqbidv 3357 . . . 4 (𝜑 → (∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun (𝑥𝑔𝑦) ↔ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun (𝑥𝑔𝑦)))
1612, 15anbi12d 633 . . 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 2801 . . . 4 (Base‘𝐴) = (Base‘𝐴)
1817isfth 17179 . . 3 (𝑓(𝐴 Faith 𝐶)𝑔 ↔ (𝑓(𝐴 Func 𝐶)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)Fun (𝑥𝑔𝑦)))
19 eqid 2801 . . . 4 (Base‘𝐵) = (Base‘𝐵)
2019isfth 17179 . . 3 (𝑓(𝐵 Faith 𝐷)𝑔 ↔ (𝑓(𝐵 Func 𝐷)𝑔 ∧ ∀𝑥 ∈ (Base‘𝐵)∀𝑦 ∈ (Base‘𝐵)Fun (𝑥𝑔𝑦)))
2116, 18, 203bitr4g 317 . 2 (𝜑 → (𝑓(𝐴 Faith 𝐶)𝑔𝑓(𝐵 Faith 𝐷)𝑔))
221, 2, 21eqbrrdiv 5635 1 (𝜑 → (𝐴 Faith 𝐶) = (𝐵 Faith 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109   class class class wbr 5033  ccnv 5522  Fun wfun 6322  cfv 6328  (class class class)co 7139  Basecbs 16478  Homf chomf 16932  compfccomf 16933   Func cfunc 17119   Faith cfth 17168
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-ixp 8449  df-cat 16934  df-cid 16935  df-homf 16936  df-comf 16937  df-func 17123  df-fth 17170
This theorem is referenced by: (None)
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