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Theorem ffthiso 17187
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 17165 . . . . . 6 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
65ssbri 5102 . . . . 5 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
74, 6syl 17 . . . 4 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
87adantr 481 . . 3 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → 𝐹(𝐶 Func 𝐷)𝐺)
9 fthmon.x . . . 4 (𝜑𝑋𝐵)
109adantr 481 . . 3 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → 𝑋𝐵)
11 fthmon.y . . . 4 (𝜑𝑌𝐵)
1211adantr 481 . . 3 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → 𝑌𝐵)
13 simpr 485 . . 3 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → 𝑅 ∈ (𝑋𝐼𝑌))
141, 2, 3, 8, 10, 12, 13funciso 17132 . 2 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
15 eqid 2818 . . . 4 (Inv‘𝐶) = (Inv‘𝐶)
16 df-br 5058 . . . . . . . 8 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
177, 16sylib 219 . . . . . . 7 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
18 funcrcl 17121 . . . . . . 7 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1917, 18syl 17 . . . . . 6 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2019simpld 495 . . . . 5 (𝜑𝐶 ∈ Cat)
2120ad3antrrr 726 . . . 4 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝐶 ∈ Cat)
229ad3antrrr 726 . . . 4 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑋𝐵)
2311ad3antrrr 726 . . . 4 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑌𝐵)
24 eqid 2818 . . . . . . . . . . 11 (Base‘𝐷) = (Base‘𝐷)
25 eqid 2818 . . . . . . . . . . 11 (Inv‘𝐷) = (Inv‘𝐷)
2619simprd 496 . . . . . . . . . . 11 (𝜑𝐷 ∈ Cat)
271, 24, 7funcf1 17124 . . . . . . . . . . . 12 (𝜑𝐹:𝐵⟶(Base‘𝐷))
2827, 9ffvelrnd 6844 . . . . . . . . . . 11 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐷))
2927, 11ffvelrnd 6844 . . . . . . . . . . 11 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐷))
3024, 25, 26, 28, 29, 3isoval 17023 . . . . . . . . . 10 (𝜑 → ((𝐹𝑋)𝐽(𝐹𝑌)) = dom ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)))
3130eleq2d 2895 . . . . . . . . 9 (𝜑 → (((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)) ↔ ((𝑋𝐺𝑌)‘𝑅) ∈ dom ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))))
3231biimpa 477 . . . . . . . 8 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ((𝑋𝐺𝑌)‘𝑅) ∈ dom ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)))
3324, 25, 26, 28, 29invfun 17022 . . . . . . . . . 10 (𝜑 → Fun ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)))
3433adantr 481 . . . . . . . . 9 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → Fun ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)))
35 funfvbrb 6813 . . . . . . . . 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 233 . . . . . . 7 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ((𝑋𝐺𝑌)‘𝑅)((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))(((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)))
3837ad2antrr 722 . . . . . 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 485 . . . . . 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 5083 . . . . 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 726 . . . . . 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 726 . . . . . 6 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑅 ∈ (𝑋𝐻𝑌))
45 simplr 765 . . . . . 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 17184 . . . . 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 258 . . . 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 17024 . . 3 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑅 ∈ (𝑋𝐼𝑌))
49 eqid 2818 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
50 ffthiso.f . . . . 5 (𝜑𝐹(𝐶 Full 𝐷)𝐺)
5150adantr 481 . . . 4 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → 𝐹(𝐶 Full 𝐷)𝐺)
5211adantr 481 . . . 4 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → 𝑌𝐵)
539adantr 481 . . . 4 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → 𝑋𝐵)
5424, 49, 3, 26, 29, 28isohom 17034 . . . . . 6 (𝜑 → ((𝐹𝑌)𝐽(𝐹𝑋)) ⊆ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
5554adantr 481 . . . . 5 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ((𝐹𝑌)𝐽(𝐹𝑋)) ⊆ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
5624, 25, 26, 28, 29, 3invf 17026 . . . . . 6 (𝜑 → ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)):((𝐹𝑋)𝐽(𝐹𝑌))⟶((𝐹𝑌)𝐽(𝐹𝑋)))
5756ffvelrnda 6843 . . . . 5 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) ∈ ((𝐹𝑌)𝐽(𝐹𝑋)))
5855, 57sseldd 3965 . . . 4 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) ∈ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
591, 49, 41, 51, 52, 53, 58fulli 17171 . . 3 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ∃𝑓 ∈ (𝑌𝐻𝑋)(((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓))
6048, 59r19.29a 3286 . 2 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → 𝑅 ∈ (𝑋𝐼𝑌))
6114, 60impbida 797 1 (𝜑 → (𝑅 ∈ (𝑋𝐼𝑌) ↔ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wss 3933  cop 4563   class class class wbr 5057  dom cdm 5548  Fun wfun 6342  cfv 6348  (class class class)co 7145  Basecbs 16471  Hom chom 16564  Catccat 16923  Invcinv 17003  Isociso 17004   Func cfunc 17112   Full cful 17160   Faith cfth 17161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-ixp 8450  df-cat 16927  df-cid 16928  df-sect 17005  df-inv 17006  df-iso 17007  df-func 17116  df-full 17162  df-fth 17163
This theorem is referenced by: (None)
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