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Theorem fucinv 16680
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 2651 . . . 4 (Sect‘𝑄) = (Sect‘𝑄)
7 eqid 2651 . . . 4 (Sect‘𝐷) = (Sect‘𝐷)
81, 2, 3, 4, 5, 6, 7fucsect 16679 . . 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 16679 . . 3 (𝜑 → (𝑉(𝐺(Sect‘𝑄)𝐹)𝑈 ↔ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))))
108, 9anbi12d 747 . 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 16667 . . 3 (𝐶 Func 𝐷) = (Base‘𝑄)
12 fucinv.i . . 3 𝐼 = (Inv‘𝑄)
13 funcrcl 16570 . . . . . 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 16677 . . 3 (𝜑𝑄 ∈ Cat)
1811, 12, 17, 4, 5, 6isinv 16467 . 2 (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈(𝐹(Sect‘𝑄)𝐺)𝑉𝑉(𝐺(Sect‘𝑄)𝐹)𝑈)))
19 eqid 2651 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
20 fucinv.j . . . . . . 7 𝐽 = (Inv‘𝐷)
2116adantr 480 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐷 ∈ Cat)
22 relfunc 16569 . . . . . . . . . 10 Rel (𝐶 Func 𝐷)
23 1st2ndbr 7261 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2422, 4, 23sylancr 696 . . . . . . . . 9 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
252, 19, 24funcf1 16573 . . . . . . . 8 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
2625ffvelrnda 6399 . . . . . . 7 ((𝜑𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
27 1st2ndbr 7261 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2822, 5, 27sylancr 696 . . . . . . . . 9 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
292, 19, 28funcf1 16573 . . . . . . . 8 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
3029ffvelrnda 6399 . . . . . . 7 ((𝜑𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
3119, 20, 21, 26, 30, 7isinv 16467 . . . . . 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 3014 . . . . 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 3093 . . . . 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 276 . . . 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 740 . . 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 1056 . . 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 1056 . . . . 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 1062 . . . . . 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 1056 . . . . . 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 264 . . . . 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 733 . . . 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 888 . . . 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 267 . . 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 303 . 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 300 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941   class class class wbr 4685  Rel wrel 5148  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Basecbs 15904  Catccat 16372  Sectcsect 16451  Invcinv 16452   Func cfunc 16561   Nat cnat 16648   FuncCat cfuc 16649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-hom 16013  df-cco 16014  df-cat 16376  df-cid 16377  df-sect 16454  df-inv 16455  df-func 16565  df-nat 16650  df-fuc 16651
This theorem is referenced by:  invfuc  16681  fuciso  16682
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