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Theorem fucinv 17231
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 2818 . . . 4 (Sect‘𝑄) = (Sect‘𝑄)
7 eqid 2818 . . . 4 (Sect‘𝐷) = (Sect‘𝐷)
81, 2, 3, 4, 5, 6, 7fucsect 17230 . . 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 17230 . . 3 (𝜑 → (𝑉(𝐺(Sect‘𝑄)𝐹)𝑈 ↔ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))))
108, 9anbi12d 630 . 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 17218 . . 3 (𝐶 Func 𝐷) = (Base‘𝑄)
12 fucinv.i . . 3 𝐼 = (Inv‘𝑄)
13 funcrcl 17121 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
144, 13syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1514simpld 495 . . . 4 (𝜑𝐶 ∈ Cat)
1614simprd 496 . . . 4 (𝜑𝐷 ∈ Cat)
171, 15, 16fuccat 17228 . . 3 (𝜑𝑄 ∈ Cat)
1811, 12, 17, 4, 5, 6isinv 17018 . 2 (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈(𝐹(Sect‘𝑄)𝐺)𝑉𝑉(𝐺(Sect‘𝑄)𝐹)𝑈)))
19 eqid 2818 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
20 fucinv.j . . . . . . 7 𝐽 = (Inv‘𝐷)
2116adantr 481 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐷 ∈ Cat)
22 relfunc 17120 . . . . . . . . . 10 Rel (𝐶 Func 𝐷)
23 1st2ndbr 7730 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2422, 4, 23sylancr 587 . . . . . . . . 9 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
252, 19, 24funcf1 17124 . . . . . . . 8 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
2625ffvelrnda 6843 . . . . . . 7 ((𝜑𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
27 1st2ndbr 7730 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2822, 5, 27sylancr 587 . . . . . . . . 9 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
292, 19, 28funcf1 17124 . . . . . . . 8 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
3029ffvelrnda 6843 . . . . . . 7 ((𝜑𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
3119, 20, 21, 26, 30, 7isinv 17018 . . . . . 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 3193 . . . . 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 3167 . . . . 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 288 . . . 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 628 . . 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 1081 . . 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 1081 . . . . 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 1090 . . . . . 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 1081 . . . . . 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 276 . . . . 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 626 . . . 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 672 . . . 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 279 . . 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 315 . 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 312 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135   class class class wbr 5057  Rel wrel 5553  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  Basecbs 16471  Catccat 16923  Sectcsect 17002  Invcinv 17003   Func cfunc 17112   Nat cnat 17199   FuncCat cfuc 17200
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  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  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-nel 3121  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-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  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-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  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-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-hom 16577  df-cco 16578  df-cat 16927  df-cid 16928  df-sect 17005  df-inv 17006  df-func 17116  df-nat 17201  df-fuc 17202
This theorem is referenced by:  invfuc  17232  fuciso  17233
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