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Theorem ffthiso 16360
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 16338 . . . . . 6 (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
65ssbri 4621 . . . . 5 (𝐹(𝐶 Faith 𝐷)𝐺𝐹(𝐶 Func 𝐷)𝐺)
74, 6syl 17 . . . 4 (𝜑𝐹(𝐶 Func 𝐷)𝐺)
87adantr 479 . . 3 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → 𝐹(𝐶 Func 𝐷)𝐺)
9 fthmon.x . . . 4 (𝜑𝑋𝐵)
109adantr 479 . . 3 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → 𝑋𝐵)
11 fthmon.y . . . 4 (𝜑𝑌𝐵)
1211adantr 479 . . 3 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → 𝑌𝐵)
13 simpr 475 . . 3 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → 𝑅 ∈ (𝑋𝐼𝑌))
141, 2, 3, 8, 10, 12, 13funciso 16305 . 2 ((𝜑𝑅 ∈ (𝑋𝐼𝑌)) → ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)))
15 eqid 2609 . . . 4 (Inv‘𝐶) = (Inv‘𝐶)
16 df-br 4578 . . . . . . . 8 (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
177, 16sylib 206 . . . . . . 7 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
18 funcrcl 16294 . . . . . . 7 (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1917, 18syl 17 . . . . . 6 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2019simpld 473 . . . . 5 (𝜑𝐶 ∈ Cat)
2120ad3antrrr 761 . . . 4 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝐶 ∈ Cat)
229ad3antrrr 761 . . . 4 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑋𝐵)
2311ad3antrrr 761 . . . 4 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑌𝐵)
24 eqid 2609 . . . . . . . . . . 11 (Base‘𝐷) = (Base‘𝐷)
25 eqid 2609 . . . . . . . . . . 11 (Inv‘𝐷) = (Inv‘𝐷)
2619simprd 477 . . . . . . . . . . 11 (𝜑𝐷 ∈ Cat)
271, 24, 7funcf1 16297 . . . . . . . . . . . 12 (𝜑𝐹:𝐵⟶(Base‘𝐷))
2827, 9ffvelrnd 6252 . . . . . . . . . . 11 (𝜑 → (𝐹𝑋) ∈ (Base‘𝐷))
2927, 11ffvelrnd 6252 . . . . . . . . . . 11 (𝜑 → (𝐹𝑌) ∈ (Base‘𝐷))
3024, 25, 26, 28, 29, 3isoval 16196 . . . . . . . . . 10 (𝜑 → ((𝐹𝑋)𝐽(𝐹𝑌)) = dom ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)))
3130eleq2d 2672 . . . . . . . . 9 (𝜑 → (((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌)) ↔ ((𝑋𝐺𝑌)‘𝑅) ∈ dom ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))))
3231biimpa 499 . . . . . . . 8 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ((𝑋𝐺𝑌)‘𝑅) ∈ dom ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)))
3324, 25, 26, 28, 29invfun 16195 . . . . . . . . . 10 (𝜑 → Fun ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)))
3433adantr 479 . . . . . . . . 9 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → Fun ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)))
35 funfvbrb 6222 . . . . . . . . 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 220 . . . . . . 7 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ((𝑋𝐺𝑌)‘𝑅)((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))(((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)))
3837ad2antrr 757 . . . . . 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 475 . . . . . 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 4603 . . . . 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 761 . . . . . 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 761 . . . . . 6 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑅 ∈ (𝑋𝐻𝑌))
45 simplr 787 . . . . . 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 16357 . . . . 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 245 . . . 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 16197 . . 3 ((((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) ∧ 𝑓 ∈ (𝑌𝐻𝑋)) ∧ (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓)) → 𝑅 ∈ (𝑋𝐼𝑌))
49 eqid 2609 . . . 4 (Hom ‘𝐷) = (Hom ‘𝐷)
50 ffthiso.f . . . . 5 (𝜑𝐹(𝐶 Full 𝐷)𝐺)
5150adantr 479 . . . 4 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → 𝐹(𝐶 Full 𝐷)𝐺)
5211adantr 479 . . . 4 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → 𝑌𝐵)
539adantr 479 . . . 4 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → 𝑋𝐵)
5424, 49, 3, 26, 29, 28isohom 16207 . . . . . 6 (𝜑 → ((𝐹𝑌)𝐽(𝐹𝑋)) ⊆ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
5554adantr 479 . . . . 5 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ((𝐹𝑌)𝐽(𝐹𝑋)) ⊆ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
5624, 25, 26, 28, 29, 3invf 16199 . . . . . 6 (𝜑 → ((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌)):((𝐹𝑋)𝐽(𝐹𝑌))⟶((𝐹𝑌)𝐽(𝐹𝑋)))
5756ffvelrnda 6251 . . . . 5 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) ∈ ((𝐹𝑌)𝐽(𝐹𝑋)))
5855, 57sseldd 3568 . . . 4 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → (((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) ∈ ((𝐹𝑌)(Hom ‘𝐷)(𝐹𝑋)))
591, 49, 41, 51, 52, 53, 58fulli 16344 . . 3 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → ∃𝑓 ∈ (𝑌𝐻𝑋)(((𝐹𝑋)(Inv‘𝐷)(𝐹𝑌))‘((𝑋𝐺𝑌)‘𝑅)) = ((𝑌𝐺𝑋)‘𝑓))
6048, 59r19.29a 3059 . 2 ((𝜑 ∧ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))) → 𝑅 ∈ (𝑋𝐼𝑌))
6114, 60impbida 872 1 (𝜑 → (𝑅 ∈ (𝑋𝐼𝑌) ↔ ((𝑋𝐺𝑌)‘𝑅) ∈ ((𝐹𝑋)𝐽(𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wss 3539  cop 4130   class class class wbr 4577  dom cdm 5027  Fun wfun 5783  cfv 5789  (class class class)co 6526  Basecbs 15643  Hom chom 15727  Catccat 16096  Invcinv 16176  Isociso 16177   Func cfunc 16285   Full cful 16333   Faith cfth 16334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-1st 7036  df-2nd 7037  df-map 7723  df-ixp 7772  df-cat 16100  df-cid 16101  df-sect 16178  df-inv 16179  df-iso 16180  df-func 16289  df-full 16335  df-fth 16336
This theorem is referenced by: (None)
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