MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fucidcl Structured version   Visualization version   GIF version

Theorem fucidcl 16606
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 16504 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
31, 2syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
43simprd 479 . . . . . 6 (𝜑𝐷 ∈ Cat)
5 eqid 2620 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
6 fucidcl.x . . . . . . 7 1 = (Id‘𝐷)
75, 6cidfn 16321 . . . . . 6 (𝐷 ∈ Cat → 1 Fn (Base‘𝐷))
84, 7syl 17 . . . . 5 (𝜑1 Fn (Base‘𝐷))
9 dffn2 6034 . . . . 5 ( 1 Fn (Base‘𝐷) ↔ 1 :(Base‘𝐷)⟶V)
108, 9sylib 208 . . . 4 (𝜑1 :(Base‘𝐷)⟶V)
11 eqid 2620 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
12 relfunc 16503 . . . . . 6 Rel (𝐶 Func 𝐷)
13 1st2ndbr 7202 . . . . . 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 16507 . . . 4 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
16 fcompt 6385 . . . 4 (( 1 :(Base‘𝐷)⟶V ∧ (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷)) → ( 1 ∘ (1st𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1st𝐹)‘𝑥))))
1710, 15, 16syl2anc 692 . . 3 (𝜑 → ( 1 ∘ (1st𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1st𝐹)‘𝑥))))
18 eqid 2620 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
194adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
2015ffvelrnda 6345 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
215, 18, 6, 19, 20catidcl 16324 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ( 1 ‘((1st𝐹)‘𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2221ralrimiva 2963 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)( 1 ‘((1st𝐹)‘𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
23 fvex 6188 . . . . 5 (Base‘𝐶) ∈ V
24 mptelixpg 7930 . . . . 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 2699 . 2 (𝜑 → ( 1 ∘ (1st𝐹)) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
284adantr 481 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝐷 ∈ Cat)
29 simpr1 1065 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
3029, 20syldan 487 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
31 eqid 2620 . . . . . 6 (comp‘𝐷) = (comp‘𝐷)
3215adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
33 simpr2 1066 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
3432, 33ffvelrnd 6346 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
35 eqid 2620 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
3614adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3711, 35, 18, 36, 29, 33funcf2 16509 . . . . . . 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 1067 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
3937, 38ffvelrnd 6346 . . . . . 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 16325 . . . . 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 16326 . . . . 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 2657 . . . 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 6262 . . . . . 6 (((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐹))‘𝑦) = ( 1 ‘((1st𝐹)‘𝑦)))
4432, 33, 43syl2anc 692 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ∘ (1st𝐹))‘𝑦) = ( 1 ‘((1st𝐹)‘𝑦)))
4544oveq1d 6650 . . . 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 6262 . . . . . 6 (((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐹))‘𝑥) = ( 1 ‘((1st𝐹)‘𝑥)))
4732, 29, 46syl2anc 692 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ∘ (1st𝐹))‘𝑥) = ( 1 ‘((1st𝐹)‘𝑥)))
4847oveq2d 6651 . . . 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 2664 . . 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 2969 . 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 16589 . 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 956 1 (𝜑 → ( 1 ∘ (1st𝐹)) ∈ (𝐹𝑁𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  cop 4174   class class class wbr 4644  cmpt 4720  ccom 5108  Rel wrel 5109   Fn wfn 5871  wf 5872  cfv 5876  (class class class)co 6635  1st c1st 7151  2nd c2nd 7152  Xcixp 7893  Basecbs 15838  Hom chom 15933  compcco 15934  Catccat 16306  Idccid 16307   Func cfunc 16495   Nat cnat 16582   FuncCat cfuc 16583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-map 7844  df-ixp 7894  df-cat 16310  df-cid 16311  df-func 16499  df-nat 16584
This theorem is referenced by:  fuclid  16607  fucrid  16608  fuccatid  16610
  Copyright terms: Public domain W3C validator