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Theorem fucinv 17243
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 2824 . . . 4 (Sect‘𝑄) = (Sect‘𝑄)
7 eqid 2824 . . . 4 (Sect‘𝐷) = (Sect‘𝐷)
81, 2, 3, 4, 5, 6, 7fucsect 17242 . . 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 17242 . . 3 (𝜑 → (𝑉(𝐺(Sect‘𝑄)𝐹)𝑈 ↔ (𝑉 ∈ (𝐺𝑁𝐹) ∧ 𝑈 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝑉𝑥)(((1st𝐺)‘𝑥)(Sect‘𝐷)((1st𝐹)‘𝑥))(𝑈𝑥))))
108, 9anbi12d 633 . 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 17230 . . 3 (𝐶 Func 𝐷) = (Base‘𝑄)
12 fucinv.i . . 3 𝐼 = (Inv‘𝑄)
13 funcrcl 17133 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
144, 13syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1514simpld 498 . . . 4 (𝜑𝐶 ∈ Cat)
1614simprd 499 . . . 4 (𝜑𝐷 ∈ Cat)
171, 15, 16fuccat 17240 . . 3 (𝜑𝑄 ∈ Cat)
1811, 12, 17, 4, 5, 6isinv 17030 . 2 (𝜑 → (𝑈(𝐹𝐼𝐺)𝑉 ↔ (𝑈(𝐹(Sect‘𝑄)𝐺)𝑉𝑉(𝐺(Sect‘𝑄)𝐹)𝑈)))
19 eqid 2824 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
20 fucinv.j . . . . . . 7 𝐽 = (Inv‘𝐷)
2116adantr 484 . . . . . . 7 ((𝜑𝑥𝐵) → 𝐷 ∈ Cat)
22 relfunc 17132 . . . . . . . . . 10 Rel (𝐶 Func 𝐷)
23 1st2ndbr 7736 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2422, 4, 23sylancr 590 . . . . . . . . 9 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
252, 19, 24funcf1 17136 . . . . . . . 8 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
2625ffvelrnda 6842 . . . . . . 7 ((𝜑𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
27 1st2ndbr 7736 . . . . . . . . . 10 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2822, 5, 27sylancr 590 . . . . . . . . 9 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
292, 19, 28funcf1 17136 . . . . . . . 8 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
3029ffvelrnda 6842 . . . . . . 7 ((𝜑𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
3119, 20, 21, 26, 30, 7isinv 17030 . . . . . 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 3191 . . . . 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 3165 . . . . 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 290 . . . 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 631 . . 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 1086 . . 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 1086 . . . . 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 1095 . . . . . 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 1086 . . . . . 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 278 . . . . 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 629 . . . 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 675 . . . 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 281 . . 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 317 . 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 314 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133   class class class wbr 5052  Rel wrel 5547  cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  Basecbs 16483  Catccat 16935  Sectcsect 17014  Invcinv 17015   Func cfunc 17124   Nat cnat 17211   FuncCat cfuc 17212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-dec 12096  df-uz 12241  df-fz 12895  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-hom 16589  df-cco 16590  df-cat 16939  df-cid 16940  df-sect 17017  df-inv 17018  df-func 17128  df-nat 17213  df-fuc 17214
This theorem is referenced by:  invfuc  17244  fuciso  17245
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