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Theorem ffthiso 17976
Description: A fully faithful functor reflects isomorphisms. Corollary 3.32 of [Adamek] p. 35. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fthmon.b 𝐵 = (Base‘𝐶)
fthmon.h 𝐻 = (Hom ‘𝐶)
fthmon.f (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
fthmon.x (𝜑𝑋𝐵)
fthmon.y (𝜑𝑌𝐵)
fthmon.r (𝜑𝑅 ∈ (𝑋𝐻𝑌))
ffthiso.f (𝜑𝐹(𝐶 Full 𝐷)𝐺)
ffthiso.s 𝐼 = (Iso‘𝐶)
ffthiso.t 𝐽 = (Iso‘𝐷)
Assertion
Ref Expression
ffthiso (𝜑 → (𝑅 ∈ (𝑋𝐼𝑌) ↔ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))))

Proof of Theorem ffthiso
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 fthmon.b . . 3 𝐵 = (Base‘𝐶)
2 ffthiso.s . . 3 𝐼 = (Iso‘𝐶)
3 ffthiso.t . . 3 𝐽 = (Iso‘𝐷)
4 fthmon.f . . . . 5 (𝜑𝐹(𝐶 Faith 𝐷)𝐺)
5 fthfunc 17954 . . . . . 6 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
65ssbri 5188 . . . . 5 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
74, 6syl 17 . . . 4 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
87adantr 480 . . 3 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → 𝐹(𝐶 Func 𝐷)𝐺)
9 fthmon.x . . . 4 (𝜑𝑋𝐵)
109adantr 480 . . 3 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → 𝑋𝐵)
11 fthmon.y . . . 4 (𝜑𝑌𝐵)
1211adantr 480 . . 3 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → 𝑌𝐵)
13 simpr 484 . . 3 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → 𝑅 ∈ (𝑋𝐼𝑌))
141, 2, 3, 8, 10, 12, 13funciso 17919 . 2 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
15 eqid 2737 . . . 4 (Inv‘𝐶) = (Inv‘𝐶)
16 df-br 5144 . . . . . . . 8 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
177, 16sylib 218 . . . . . . 7 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
18 funcrcl 17908 . . . . . . 7 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1917, 18syl 17 . . . . . 6 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2019simpld 494 . . . . 5 (𝜑𝐶 ∈ Cat)
2120ad3antrrr 730 . . . 4 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝐶 ∈ Cat)
229ad3antrrr 730 . . . 4 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑋𝐵)
2311ad3antrrr 730 . . . 4 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑌𝐵)
24 eqid 2737 . . . . . . . . . . 11 (Base‘𝐷) = (Base‘𝐷)
25 eqid 2737 . . . . . . . . . . 11 (Inv‘𝐷) = (Inv‘𝐷)
2619simprd 495 . . . . . . . . . . 11 (𝜑𝐷 ∈ Cat)
271, 24, 7funcf1 17911 . . . . . . . . . . . 12 (𝜑𝐹:𝐵⟶(Base‘𝐷))
2827, 9ffvelcdmd 7105 . . . . . . . . . . 11 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐷))
2927, 11ffvelcdmd 7105 . . . . . . . . . . 11 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐷))
3024, 25, 26, 28, 29, 3isoval 17809 . . . . . . . . . 10 (𝜑 → ((𝐹𝑋)𝐽(𝐹𝑌)) = dom ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)))
3130eleq2d 2827 . . . . . . . . 9 (𝜑 → (((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)) ↔ ((𝑋𝐺𝑌)‘𝑅) ∈ dom ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))))
3231biimpa 476 . . . . . . . 8 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ((𝑋𝐺𝑌)‘𝑅) ∈ dom ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)))
3324, 25, 26, 28, 29invfun 17808 . . . . . . . . . 10 (𝜑 → Fun ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)))
3433adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → Fun ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)))
35 funfvbrb 7071 . . . . . . . . 9 (Fun ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)) → (((𝑋𝐺𝑌)‘𝑅) ∈ dom ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)) ↔ ((𝑋𝐺𝑌)‘𝑅)((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))(((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅))))
3634, 35syl 17 . . . . . . . 8 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → (((𝑋𝐺𝑌)‘𝑅) ∈ dom ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)) ↔ ((𝑋𝐺𝑌)‘𝑅)((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))(((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅))))
3732, 36mpbid 232 . . . . . . 7 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ((𝑋𝐺𝑌)‘𝑅)((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))(((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)))
3837ad2antrr 726 . . . . . 6 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → ((𝑋𝐺𝑌)‘𝑅)((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))(((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)))
39 simpr 484 . . . . . 6 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓))
4038, 39breqtrd 5169 . . . . 5 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → ((𝑋𝐺𝑌)‘𝑅)((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))((𝑌𝐺𝑋)‘𝑓))
41 fthmon.h . . . . . 6 𝐻 = (Hom ‘𝐶)
424ad3antrrr 730 . . . . . 6 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝐹(𝐶 Faith 𝐷)𝐺)
43 fthmon.r . . . . . . 7 (𝜑𝑅 ∈ (𝑋𝐻𝑌))
4443ad3antrrr 730 . . . . . 6 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑅 ∈ (𝑋𝐻𝑌))
45 simplr 769 . . . . . 6 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑓 ∈ (𝑌𝐻𝑋))
461, 41, 42, 22, 23, 44, 45, 15, 25fthinv 17973 . . . . 5 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → (𝑅(𝑋(Inv‘𝐶)𝑌)𝑓 ↔ ((𝑋𝐺𝑌)‘𝑅)((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))((𝑌𝐺𝑋)‘𝑓)))
4740, 46mpbird 257 . . . 4 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑅(𝑋(Inv‘𝐶)𝑌)𝑓)
481, 15, 21, 22, 23, 2, 47inviso1 17810 . . 3 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑅 ∈ (𝑋𝐼𝑌))
49 eqid 2737 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
50 ffthiso.f . . . . 5 (𝜑𝐹(𝐶 Full 𝐷)𝐺)
5150adantr 480 . . . 4 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → 𝐹(𝐶 Full 𝐷)𝐺)
5211adantr 480 . . . 4 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → 𝑌𝐵)
539adantr 480 . . . 4 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → 𝑋𝐵)
5424, 49, 3, 26, 29, 28isohom 17820 . . . . . 6 (𝜑 → ((𝐹𝑌)𝐽(𝐹𝑋)) ⊆ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
5554adantr 480 . . . . 5 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ((𝐹𝑌)𝐽(𝐹𝑋)) ⊆ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
5624, 25, 26, 28, 29, 3invf 17812 . . . . . 6 (𝜑 → ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)):((𝐹𝑋)𝐽(𝐹𝑌))⟶((𝐹𝑌)𝐽(𝐹𝑋)))
5756ffvelcdmda 7104 . . . . 5 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) ∈ ((𝐹𝑌)𝐽(𝐹𝑋)))
5855, 57sseldd 3984 . . . 4 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) ∈ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
591, 49, 41, 51, 52, 53, 58fulli 17960 . . 3 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ∃𝑓 ∈ (𝑌𝐻𝑋)(((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓))
6048, 59r19.29a 3162 . 2 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → 𝑅 ∈ (𝑋𝐼𝑌))
6114, 60impbida 801 1 (𝜑 → (𝑅 ∈ (𝑋𝐼𝑌) ↔ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951  cop 4632   class class class wbr 5143  dom cdm 5685  Fun wfun 6555  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  Catccat 17707  Invcinv 17789  Isociso 17790   Func cfunc 17899   Full cful 17949   Faith cfth 17950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ixp 8938  df-cat 17711  df-cid 17712  df-sect 17791  df-inv 17792  df-iso 17793  df-func 17903  df-full 17951  df-fth 17952
This theorem is referenced by: (None)
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