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Theorem fucidcl 16846
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 16744 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
31, 2syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
43simprd 482 . . . . . 6 (𝜑𝐷 ∈ Cat)
5 eqid 2760 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
6 fucidcl.x . . . . . . 7 1 = (Id‘𝐷)
75, 6cidfn 16561 . . . . . 6 (𝐷 ∈ Cat → 1 Fn (Base‘𝐷))
84, 7syl 17 . . . . 5 (𝜑1 Fn (Base‘𝐷))
9 dffn2 6208 . . . . 5 ( 1 Fn (Base‘𝐷) ↔ 1 :(Base‘𝐷)⟶V)
108, 9sylib 208 . . . 4 (𝜑1 :(Base‘𝐷)⟶V)
11 eqid 2760 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
12 relfunc 16743 . . . . . 6 Rel (𝐶 Func 𝐷)
13 1st2ndbr 7385 . . . . . 6 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1412, 1, 13sylancr 698 . . . . 5 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1511, 5, 14funcf1 16747 . . . 4 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
16 fcompt 6564 . . . 4 (( 1 :(Base‘𝐷)⟶V ∧ (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷)) → ( 1 ∘ (1st𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1st𝐹)‘𝑥))))
1710, 15, 16syl2anc 696 . . 3 (𝜑 → ( 1 ∘ (1st𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1st𝐹)‘𝑥))))
18 eqid 2760 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
194adantr 472 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
2015ffvelrnda 6523 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
215, 18, 6, 19, 20catidcl 16564 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ( 1 ‘((1st𝐹)‘𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2221ralrimiva 3104 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)( 1 ‘((1st𝐹)‘𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
23 fvex 6363 . . . . 5 (Base‘𝐶) ∈ V
24 mptelixpg 8113 . . . . 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 224 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1st𝐹)‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2717, 26eqeltrd 2839 . 2 (𝜑 → ( 1 ∘ (1st𝐹)) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
284adantr 472 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝐷 ∈ Cat)
29 simpr1 1234 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
3029, 20syldan 488 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
31 eqid 2760 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
3215adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
33 simpr2 1236 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
3432, 33ffvelrnd 6524 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
35 eqid 2760 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
3614adantr 472 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3711, 35, 18, 36, 29, 33funcf2 16749 . . . . . . 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 1238 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
3937, 38ffvelrnd 6524 . . . . . 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 16565 . . . . 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 16566 . . . . 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 2797 . . . 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 6438 . . . . . 6 (((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐹))‘𝑦) = ( 1 ‘((1st𝐹)‘𝑦)))
4432, 33, 43syl2anc 696 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ∘ (1st𝐹))‘𝑦) = ( 1 ‘((1st𝐹)‘𝑦)))
4544oveq1d 6829 . . . 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 6438 . . . . . 6 (((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐹))‘𝑥) = ( 1 ‘((1st𝐹)‘𝑥)))
4732, 29, 46syl2anc 696 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ∘ (1st𝐹))‘𝑥) = ( 1 ‘((1st𝐹)‘𝑥)))
4847oveq2d 6830 . . . 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 2804 . . 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 3110 . 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 16829 . 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 995 1 (𝜑 → ( 1 ∘ (1st𝐹)) ∈ (𝐹𝑁𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cop 4327   class class class wbr 4804  cmpt 4881  ccom 5270  Rel wrel 5271   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6814  1st c1st 7332  2nd c2nd 7333  Xcixp 8076  Basecbs 16079  Hom chom 16174  compcco 16175  Catccat 16546  Idccid 16547   Func cfunc 16735   Nat cnat 16822   FuncCat cfuc 16823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027  df-ixp 8077  df-cat 16550  df-cid 16551  df-func 16739  df-nat 16824
This theorem is referenced by:  fuclid  16847  fucrid  16848  fuccatid  16850
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