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Theorem ffthiso 17983
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 17961 . . . . . 6 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
65ssbri 5193 . . . . 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 17925 . 2 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
15 eqid 2735 . . . 4 (Inv‘𝐶) = (Inv‘𝐶)
16 df-br 5149 . . . . . . . 8 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
177, 16sylib 218 . . . . . . 7 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
18 funcrcl 17914 . . . . . . 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 2735 . . . . . . . . . . 11 (Base‘𝐷) = (Base‘𝐷)
25 eqid 2735 . . . . . . . . . . 11 (Inv‘𝐷) = (Inv‘𝐷)
2619simprd 495 . . . . . . . . . . 11 (𝜑𝐷 ∈ Cat)
271, 24, 7funcf1 17917 . . . . . . . . . . . 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 17813 . . . . . . . . . 10 (𝜑 → ((𝐹𝑋)𝐽(𝐹𝑌)) = dom ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)))
3130eleq2d 2825 . . . . . . . . 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 17812 . . . . . . . . . 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 5174 . . . . 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 17980 . . . . 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 17814 . . 3 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑅 ∈ (𝑋𝐼𝑌))
49 eqid 2735 . . . 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 17824 . . . . . 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 17816 . . . . . 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 3996 . . . 4 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) ∈ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
591, 49, 41, 51, 52, 53, 58fulli 17967 . . 3 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ∃𝑓 ∈ (𝑌𝐻𝑋)(((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓))
6048, 59r19.29a 3160 . 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 1537  wcel 2106  wss 3963  cop 4637   class class class wbr 5148  dom cdm 5689  Fun wfun 6557  cfv 6563  (class class class)co 7431  Basecbs 17245  Hom chom 17309  Catccat 17709  Invcinv 17793  Isociso 17794   Func cfunc 17905   Full cful 17956   Faith cfth 17957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-ixp 8937  df-cat 17713  df-cid 17714  df-sect 17795  df-inv 17796  df-iso 17797  df-func 17909  df-full 17958  df-fth 17959
This theorem is referenced by: (None)
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