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Theorem fuciso 17115
Description: A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (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))
fuciso.i 𝐼 = (Iso‘𝑄)
fuciso.j 𝐽 = (Iso‘𝐷)
Assertion
Ref Expression
fuciso (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝑥,𝐼   𝑥,𝐹   𝑥,𝐺   𝑥,𝐽   𝑥,𝑁   𝜑,𝑥   𝑥,𝑄

Proof of Theorem fuciso
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fuciso.q . . . . . 6 𝑄 = (𝐶 FuncCat 𝐷)
21fucbas 17100 . . . . 5 (𝐶 Func 𝐷) = (Base‘𝑄)
3 fuciso.n . . . . . 6 𝑁 = (𝐶 Nat 𝐷)
41, 3fuchom 17101 . . . . 5 𝑁 = (Hom ‘𝑄)
5 fuciso.i . . . . 5 𝐼 = (Iso‘𝑄)
6 fuciso.f . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
7 funcrcl 17003 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
86, 7syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
98simpld 487 . . . . . 6 (𝜑𝐶 ∈ Cat)
108simprd 488 . . . . . 6 (𝜑𝐷 ∈ Cat)
111, 9, 10fuccat 17110 . . . . 5 (𝜑𝑄 ∈ Cat)
12 fuciso.g . . . . 5 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
132, 4, 5, 11, 6, 12isohom 16916 . . . 4 (𝜑 → (𝐹𝐼𝐺) ⊆ (𝐹𝑁𝐺))
1413sselda 3852 . . 3 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴 ∈ (𝐹𝑁𝐺))
15 eqid 2772 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
16 eqid 2772 . . . . 5 (Inv‘𝐷) = (Inv‘𝐷)
1710ad2antrr 713 . . . . 5 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → 𝐷 ∈ Cat)
18 fuciso.b . . . . . . . 8 𝐵 = (Base‘𝐶)
19 relfunc 17002 . . . . . . . . 9 Rel (𝐶 Func 𝐷)
20 1st2ndbr 7551 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2119, 6, 20sylancr 578 . . . . . . . 8 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
2218, 15, 21funcf1 17006 . . . . . . 7 (𝜑 → (1st𝐹):𝐵⟶(Base‘𝐷))
2322adantr 473 . . . . . 6 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → (1st𝐹):𝐵⟶(Base‘𝐷))
2423ffvelrnda 6674 . . . . 5 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
25 1st2ndbr 7551 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2619, 12, 25sylancr 578 . . . . . . . 8 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
2718, 15, 26funcf1 17006 . . . . . . 7 (𝜑 → (1st𝐺):𝐵⟶(Base‘𝐷))
2827adantr 473 . . . . . 6 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → (1st𝐺):𝐵⟶(Base‘𝐷))
2928ffvelrnda 6674 . . . . 5 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
30 fuciso.j . . . . 5 𝐽 = (Iso‘𝐷)
31 eqid 2772 . . . . . . . . . . . 12 (Inv‘𝑄) = (Inv‘𝑄)
322, 31, 11, 6, 12, 5isoval 16905 . . . . . . . . . . 11 (𝜑 → (𝐹𝐼𝐺) = dom (𝐹(Inv‘𝑄)𝐺))
3332eleq2d 2845 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺)))
342, 31, 11, 6, 12invfun 16904 . . . . . . . . . . 11 (𝜑 → Fun (𝐹(Inv‘𝑄)𝐺))
35 funfvbrb 6644 . . . . . . . . . . 11 (Fun (𝐹(Inv‘𝑄)𝐺) → (𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺) ↔ 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴)))
3634, 35syl 17 . . . . . . . . . 10 (𝜑 → (𝐴 ∈ dom (𝐹(Inv‘𝑄)𝐺) ↔ 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴)))
3733, 36bitrd 271 . . . . . . . . 9 (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴)))
3837biimpa 469 . . . . . . . 8 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → 𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴))
391, 18, 3, 6, 12, 31, 16fucinv 17113 . . . . . . . . 9 (𝜑 → (𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ((𝐹(Inv‘𝑄)𝐺)‘𝐴) ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝐴𝑥)(((1st𝐹)‘𝑥)(Inv‘𝐷)((1st𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))))
4039adantr 473 . . . . . . . 8 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → (𝐴(𝐹(Inv‘𝑄)𝐺)((𝐹(Inv‘𝑄)𝐺)‘𝐴) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ((𝐹(Inv‘𝑄)𝐺)‘𝐴) ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝐴𝑥)(((1st𝐹)‘𝑥)(Inv‘𝐷)((1st𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))))
4138, 40mpbid 224 . . . . . . 7 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → (𝐴 ∈ (𝐹𝑁𝐺) ∧ ((𝐹(Inv‘𝑄)𝐺)‘𝐴) ∈ (𝐺𝑁𝐹) ∧ ∀𝑥𝐵 (𝐴𝑥)(((1st𝐹)‘𝑥)(Inv‘𝐷)((1st𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥)))
4241simp3d 1124 . . . . . 6 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → ∀𝑥𝐵 (𝐴𝑥)(((1st𝐹)‘𝑥)(Inv‘𝐷)((1st𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))
4342r19.21bi 3152 . . . . 5 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → (𝐴𝑥)(((1st𝐹)‘𝑥)(Inv‘𝐷)((1st𝐺)‘𝑥))(((𝐹(Inv‘𝑄)𝐺)‘𝐴)‘𝑥))
4415, 16, 17, 24, 29, 30, 43inviso1 16906 . . . 4 (((𝜑𝐴 ∈ (𝐹𝐼𝐺)) ∧ 𝑥𝐵) → (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))
4544ralrimiva 3126 . . 3 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))
4614, 45jca 504 . 2 ((𝜑𝐴 ∈ (𝐹𝐼𝐺)) → (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥))))
4711adantr 473 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝑄 ∈ Cat)
486adantr 473 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐹 ∈ (𝐶 Func 𝐷))
4912adantr 473 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐺 ∈ (𝐶 Func 𝐷))
50 simprl 758 . . . 4 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝑁𝐺))
5110ad2antrr 713 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → 𝐷 ∈ Cat)
5222adantr 473 . . . . . 6 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → (1st𝐹):𝐵⟶(Base‘𝐷))
5352ffvelrnda 6674 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
5427adantr 473 . . . . . 6 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → (1st𝐺):𝐵⟶(Base‘𝐷))
5554ffvelrnda 6674 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐷))
56 simprr 760 . . . . . 6 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))
57 fveq2 6496 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥) = (𝐴𝑦))
58 fveq2 6496 . . . . . . . . 9 (𝑥 = 𝑦 → ((1st𝐹)‘𝑥) = ((1st𝐹)‘𝑦))
59 fveq2 6496 . . . . . . . . 9 (𝑥 = 𝑦 → ((1st𝐺)‘𝑥) = ((1st𝐺)‘𝑦))
6058, 59oveq12d 6992 . . . . . . . 8 (𝑥 = 𝑦 → (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) = (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
6157, 60eleq12d 2854 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ↔ (𝐴𝑦) ∈ (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦))))
6261rspccva 3528 . . . . . 6 ((∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)) ∧ 𝑦𝐵) → (𝐴𝑦) ∈ (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
6356, 62sylan 572 . . . . 5 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → (𝐴𝑦) ∈ (((1st𝐹)‘𝑦)𝐽((1st𝐺)‘𝑦)))
6415, 30, 16, 51, 53, 55, 63invisoinvr 16931 . . . 4 (((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) ∧ 𝑦𝐵) → (𝐴𝑦)(((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦))((((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦))‘(𝐴𝑦)))
651, 18, 3, 48, 49, 31, 16, 50, 64invfuc 17114 . . 3 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐴(𝐹(Inv‘𝑄)𝐺)(𝑦𝐵 ↦ ((((1st𝐹)‘𝑦)(Inv‘𝐷)((1st𝐺)‘𝑦))‘(𝐴𝑦))))
662, 31, 47, 48, 49, 5, 65inviso1 16906 . 2 ((𝜑 ∧ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))) → 𝐴 ∈ (𝐹𝐼𝐺))
6746, 66impbida 788 1 (𝜑 → (𝐴 ∈ (𝐹𝐼𝐺) ↔ (𝐴 ∈ (𝐹𝑁𝐺) ∧ ∀𝑥𝐵 (𝐴𝑥) ∈ (((1st𝐹)‘𝑥)𝐽((1st𝐺)‘𝑥)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wral 3082   class class class wbr 4925  cmpt 5004  dom cdm 5403  Rel wrel 5408  Fun wfun 6179  wf 6181  cfv 6185  (class class class)co 6974  1st c1st 7497  2nd c2nd 7498  Basecbs 16337  Catccat 16805  Invcinv 16885  Isociso 16886   Func cfunc 16994   Nat cnat 17081   FuncCat cfuc 17082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-oadd 7907  df-er 8087  df-map 8206  df-ixp 8258  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-nn 11438  df-2 11501  df-3 11502  df-4 11503  df-5 11504  df-6 11505  df-7 11506  df-8 11507  df-9 11508  df-n0 11706  df-z 11792  df-dec 11910  df-uz 12057  df-fz 12707  df-struct 16339  df-ndx 16340  df-slot 16341  df-base 16343  df-hom 16443  df-cco 16444  df-cat 16809  df-cid 16810  df-sect 16887  df-inv 16888  df-iso 16889  df-func 16998  df-nat 17083  df-fuc 17084
This theorem is referenced by:  yonffthlem  17402
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