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Theorem fucinv 16404
 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 2609 . . . 4 (Sect‘𝑄) = (Sect‘𝑄)
7 eqid 2609 . . . 4 (Sect‘𝐷) = (Sect‘𝐷)
81, 2, 3, 4, 5, 6, 7fucsect 16403 . . 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 16403 . . 3 (𝜑 → (𝑉(𝐺(Sect‘𝑄)𝐹)𝑈 ↔ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))))
108, 9anbi12d 742 . 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 16391 . . 3 (𝐶 Func 𝐷) = (Base‘𝑄)
12 fucinv.i . . 3 𝐼 = (Inv‘𝑄)
13 funcrcl 16294 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
144, 13syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1514simpld 473 . . . 4 (𝜑𝐶 ∈ Cat)
1614simprd 477 . . . 4 (𝜑𝐷 ∈ Cat)
171, 15, 16fuccat 16401 . . 3 (𝜑𝑄 ∈ Cat)
1811, 12, 17, 4, 5, 6isinv 16191 . 2 (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈(𝐹(Sect‘𝑄)𝐺)𝑉𝑉(𝐺(Sect‘𝑄)𝐹)𝑈)))
19 eqid 2609 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
20 fucinv.j . . . . . . 7 𝐽 = (Inv‘𝐷)
2116adantr 479 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐷 ∈ Cat)
22 relfunc 16293 . . . . . . . . . 10 Rel (𝐶 Func 𝐷)
23 1st2ndbr 7085 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2422, 4, 23sylancr 693 . . . . . . . . 9 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
252, 19, 24funcf1 16297 . . . . . . . 8 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
2625ffvelrnda 6251 . . . . . . 7 ((𝜑𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
27 1st2ndbr 7085 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2822, 5, 27sylancr 693 . . . . . . . . 9 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
292, 19, 28funcf1 16297 . . . . . . . 8 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
3029ffvelrnda 6251 . . . . . . 7 ((𝜑𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
3119, 20, 21, 26, 30, 7isinv 16191 . . . . . 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 2967 . . . . 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 3045 . . . . 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 274 . . . 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 735 . . 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 1032 . . 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 1032 . . . . 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 1037 . . . . . 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 1032 . . . . . 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 262 . . . . 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 728 . . . 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 866 . . . 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 265 . . 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 301 . 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 298 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 194   ∧ wa 382   ∧ w3a 1030   = wceq 1474   ∈ wcel 1976  ∀wral 2895   class class class wbr 4577  Rel wrel 5032  ‘cfv 5789  (class class class)co 6526  1st c1st 7034  2nd c2nd 7035  Basecbs 15643  Catccat 16096  Sectcsect 16175  Invcinv 16176   Func cfunc 16285   Nat cnat 16372   FuncCat cfuc 16373 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  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  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-nel 2782  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-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  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-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  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-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10870  df-2 10928  df-3 10929  df-4 10930  df-5 10931  df-6 10932  df-7 10933  df-8 10934  df-9 10935  df-n0 11142  df-z 11213  df-dec 11328  df-uz 11522  df-fz 12155  df-struct 15645  df-ndx 15646  df-slot 15647  df-base 15648  df-hom 15741  df-cco 15742  df-cat 16100  df-cid 16101  df-sect 16178  df-inv 16179  df-func 16289  df-nat 16374  df-fuc 16375 This theorem is referenced by:  invfuc  16405  fuciso  16406
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