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Theorem fucinv 16456
 Description: Two natural transformations are inverses of each other iff all the components are inverse. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
fuciso.q 𝑄 = (𝐶 FuncCat 𝐷)
fuciso.b 𝐵 = (Base‘𝐶)
fuciso.n 𝑁 = (𝐶 Nat 𝐷)
fuciso.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
fuciso.g (𝜑𝐺 ∈ (𝐶 Func 𝐷))
fucinv.i 𝐼 = (Inv‘𝑄)
fucinv.j 𝐽 = (Inv‘𝐷)
Assertion
Ref Expression
fucinv (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))(𝑉𝑥))))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝑥,𝐼   𝑥,𝐹   𝑥,𝐺   𝑥,𝐽   𝑥,𝑁   𝑥,𝑉   𝜑,𝑥   𝑥,𝑄   𝑥,𝑈

Proof of Theorem fucinv
StepHypRef Expression
1 fuciso.q . . . 4 𝑄 = (𝐶 FuncCat 𝐷)
2 fuciso.b . . . 4 𝐵 = (Base‘𝐶)
3 fuciso.n . . . 4 𝑁 = (𝐶 Nat 𝐷)
4 fuciso.f . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
5 fuciso.g . . . 4 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
6 eqid 2610 . . . 4 (Sect‘𝑄) = (Sect‘𝑄)
7 eqid 2610 . . . 4 (Sect‘𝐷) = (Sect‘𝐷)
81, 2, 3, 4, 5, 6, 7fucsect 16455 . . 3 (𝜑 → (𝑈(𝐹(Sect‘𝑄)𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥))))
91, 2, 3, 5, 4, 6, 7fucsect 16455 . . 3 (𝜑 → (𝑉(𝐺(Sect‘𝑄)𝐹)𝑈 ↔ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))))
108, 9anbi12d 743 . 2 (𝜑 → ((𝑈(𝐹(Sect‘𝑄)𝐺)𝑉𝑉(𝐺(Sect‘𝑄)𝐹)𝑈) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥)) ∧ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)))))
111fucbas 16443 . . 3 (𝐶 Func 𝐷) = (Base‘𝑄)
12 fucinv.i . . 3 𝐼 = (Inv‘𝑄)
13 funcrcl 16346 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
144, 13syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1514simpld 474 . . . 4 (𝜑𝐶 ∈ Cat)
1614simprd 478 . . . 4 (𝜑𝐷 ∈ Cat)
171, 15, 16fuccat 16453 . . 3 (𝜑𝑄 ∈ Cat)
1811, 12, 17, 4, 5, 6isinv 16243 . 2 (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈(𝐹(Sect‘𝑄)𝐺)𝑉𝑉(𝐺(Sect‘𝑄)𝐹)𝑈)))
19 eqid 2610 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
20 fucinv.j . . . . . . 7 𝐽 = (Inv‘𝐷)
2116adantr 480 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐷 ∈ Cat)
22 relfunc 16345 . . . . . . . . . 10 Rel (𝐶 Func 𝐷)
23 1st2ndbr 7108 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2422, 4, 23sylancr 694 . . . . . . . . 9 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
252, 19, 24funcf1 16349 . . . . . . . 8 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
2625ffvelrnda 6267 . . . . . . 7 ((𝜑𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
27 1st2ndbr 7108 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2822, 5, 27sylancr 694 . . . . . . . . 9 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
292, 19, 28funcf1 16349 . . . . . . . 8 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
3029ffvelrnda 6267 . . . . . . 7 ((𝜑𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
3119, 20, 21, 26, 30, 7isinv 16243 . . . . . 6 ((𝜑𝑥𝐵) → ((𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))(𝑉𝑥) ↔ ((𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥) ∧ (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))))
3231ralbidva 2968 . . . . 5 (𝜑 → (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))(𝑉𝑥) ↔ ∀𝑥𝐵 ((𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥) ∧ (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))))
33 r19.26 3046 . . . . 5 (∀𝑥𝐵 ((𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥) ∧ (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) ↔ (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)))
3432, 33syl6bb 275 . . . 4 (𝜑 → (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))(𝑉𝑥) ↔ (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))))
3534anbi2d 736 . . 3 (𝜑 → (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))(𝑉𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)))))
36 df-3an 1033 . . 3 ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))(𝑉𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))(𝑉𝑥)))
37 df-3an 1033 . . . . 5 ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥)))
38 3ancoma 1038 . . . . . 6 ((𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)))
39 df-3an 1033 . . . . . 6 ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)))
4038, 39bitri 263 . . . . 5 ((𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)))
4137, 40anbi12i 729 . . . 4 (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥)) ∧ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))) ↔ (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥)) ∧ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))))
42 anandi 867 . . . 4 (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))) ↔ (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥)) ∧ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))))
4341, 42bitr4i 266 . . 3 (((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥)) ∧ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹)) ∧ (∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))))
4435, 36, 433bitr4g 302 . 2 (𝜑 → ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))(𝑉𝑥)) ↔ ((𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)(Sect‘𝐷)((1st𝐺)‘𝑥))(𝑉𝑥)) ∧ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥)))))
4510, 18, 443bitr4d 299 1 (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈 ∈ (𝐹𝑁𝐺) ∧ 𝑉 ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝑈𝑥)(((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))(𝑉𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  Catccat 16148  Sectcsect 16227  Invcinv 16228   Func cfunc 16337   Nat cnat 16424   FuncCat cfuc 16425 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-sect 16230  df-inv 16231  df-func 16341  df-nat 16426  df-fuc 16427 This theorem is referenced by:  invfuc  16457  fuciso  16458
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