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Theorem fucidcl 16448
 Description: The identity natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fucidcl.q 𝑄 = (𝐶 FuncCat 𝐷)
fucidcl.n 𝑁 = (𝐶 Nat 𝐷)
fucidcl.x 1 = (Id‘𝐷)
fucidcl.f (𝜑𝐹 ∈ (𝐶 Func 𝐷))
Assertion
Ref Expression
fucidcl (𝜑 → ( 1 ∘ (1st𝐹)) ∈ (𝐹𝑁𝐹))

Proof of Theorem fucidcl
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucidcl.f . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
2 funcrcl 16346 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
31, 2syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
43simprd 478 . . . . . 6 (𝜑𝐷 ∈ Cat)
5 eqid 2610 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
6 fucidcl.x . . . . . . 7 1 = (Id‘𝐷)
75, 6cidfn 16163 . . . . . 6 (𝐷 ∈ Cat → 1 Fn (Base‘𝐷))
84, 7syl 17 . . . . 5 (𝜑1 Fn (Base‘𝐷))
9 dffn2 5960 . . . . 5 ( 1 Fn (Base‘𝐷) ↔ 1 :(Base‘𝐷)⟶V)
108, 9sylib 207 . . . 4 (𝜑1 :(Base‘𝐷)⟶V)
11 eqid 2610 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
12 relfunc 16345 . . . . . 6 Rel (𝐶 Func 𝐷)
13 1st2ndbr 7108 . . . . . 6 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1412, 1, 13sylancr 694 . . . . 5 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1511, 5, 14funcf1 16349 . . . 4 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
16 fcompt 6306 . . . 4 (( 1 :(Base‘𝐷)⟶V ∧ (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷)) → ( 1 ∘ (1st𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1st𝐹)‘𝑥))))
1710, 15, 16syl2anc 691 . . 3 (𝜑 → ( 1 ∘ (1st𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1st𝐹)‘𝑥))))
18 eqid 2610 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
194adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
2015ffvelrnda 6267 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
215, 18, 6, 19, 20catidcl 16166 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ( 1 ‘((1st𝐹)‘𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2221ralrimiva 2949 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)( 1 ‘((1st𝐹)‘𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
23 fvex 6113 . . . . 5 (Base‘𝐶) ∈ V
24 mptelixpg 7831 . . . . 5 ((Base‘𝐶) ∈ V → ((𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1st𝐹)‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐶)( 1 ‘((1st𝐹)‘𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥))))
2523, 24ax-mp 5 . . . 4 ((𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1st𝐹)‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) ↔ ∀𝑥 ∈ (Base‘𝐶)( 1 ‘((1st𝐹)‘𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2622, 25sylibr 223 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1st𝐹)‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2717, 26eqeltrd 2688 . 2 (𝜑 → ( 1 ∘ (1st𝐹)) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
284adantr 480 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝐷 ∈ Cat)
29 simpr1 1060 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
3029, 20syldan 486 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
31 eqid 2610 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
3215adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
33 simpr2 1061 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
3432, 33ffvelrnd 6268 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
35 eqid 2610 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
3614adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3711, 35, 18, 36, 29, 33funcf2 16351 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
38 simpr3 1062 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
3937, 38ffvelrnd 6268 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
405, 18, 6, 28, 30, 31, 34, 39catlid 16167 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ‘((1st𝐹)‘𝑦))(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)) = ((𝑥(2nd𝐹)𝑦)‘𝑓))
415, 18, 6, 28, 30, 31, 34, 39catrid 16168 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑥(2nd𝐹)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑦))( 1 ‘((1st𝐹)‘𝑥))) = ((𝑥(2nd𝐹)𝑦)‘𝑓))
4240, 41eqtr4d 2647 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ‘((1st𝐹)‘𝑦))(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)) = (((𝑥(2nd𝐹)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑦))( 1 ‘((1st𝐹)‘𝑥))))
43 fvco3 6185 . . . . . 6 (((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐹))‘𝑦) = ( 1 ‘((1st𝐹)‘𝑦)))
4432, 33, 43syl2anc 691 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ∘ (1st𝐹))‘𝑦) = ( 1 ‘((1st𝐹)‘𝑦)))
4544oveq1d 6564 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((( 1 ∘ (1st𝐹))‘𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)) = (( 1 ‘((1st𝐹)‘𝑦))(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)))
46 fvco3 6185 . . . . . 6 (((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐹))‘𝑥) = ( 1 ‘((1st𝐹)‘𝑥)))
4732, 29, 46syl2anc 691 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ∘ (1st𝐹))‘𝑥) = ( 1 ‘((1st𝐹)‘𝑥)))
4847oveq2d 6565 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (((𝑥(2nd𝐹)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑦))(( 1 ∘ (1st𝐹))‘𝑥)) = (((𝑥(2nd𝐹)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑦))( 1 ‘((1st𝐹)‘𝑥))))
4942, 45, 483eqtr4d 2654 . . 3 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((( 1 ∘ (1st𝐹))‘𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)) = (((𝑥(2nd𝐹)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑦))(( 1 ∘ (1st𝐹))‘𝑥)))
5049ralrimivvva 2955 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)((( 1 ∘ (1st𝐹))‘𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)) = (((𝑥(2nd𝐹)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑦))(( 1 ∘ (1st𝐹))‘𝑥)))
51 fucidcl.n . . 3 𝑁 = (𝐶 Nat 𝐷)
5251, 11, 35, 18, 31, 1, 1isnat2 16431 . 2 (𝜑 → (( 1 ∘ (1st𝐹)) ∈ (𝐹𝑁𝐹) ↔ (( 1 ∘ (1st𝐹)) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)) ∧ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)((( 1 ∘ (1st𝐹))‘𝑦)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑦))((𝑥(2nd𝐹)𝑦)‘𝑓)) = (((𝑥(2nd𝐹)𝑦)‘𝑓)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐹)‘𝑦))(( 1 ∘ (1st𝐹))‘𝑥)))))
5327, 50, 52mpbir2and 959 1 (𝜑 → ( 1 ∘ (1st𝐹)) ∈ (𝐹𝑁𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   ∘ ccom 5042  Rel wrel 5043   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Xcixp 7794  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149   Func cfunc 16337   Nat cnat 16424   FuncCat cfuc 16425 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-ixp 7795  df-cat 16152  df-cid 16153  df-func 16341  df-nat 16426 This theorem is referenced by:  fuclid  16449  fucrid  16450  fuccatid  16452
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