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Theorem fucidcl 17854
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 17749 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
31, 2syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
43simprd 496 . . . . . 6 (𝜑𝐷 ∈ Cat)
5 eqid 2736 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
6 fucidcl.x . . . . . . 7 1 = (Id‘𝐷)
75, 6cidfn 17559 . . . . . 6 (𝐷 ∈ Cat → 1 Fn (Base‘𝐷))
84, 7syl 17 . . . . 5 (𝜑1 Fn (Base‘𝐷))
9 dffn2 6670 . . . . 5 ( 1 Fn (Base‘𝐷) ↔ 1 :(Base‘𝐷)⟶V)
108, 9sylib 217 . . . 4 (𝜑1 :(Base‘𝐷)⟶V)
11 eqid 2736 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
12 relfunc 17748 . . . . . 6 Rel (𝐶 Func 𝐷)
13 1st2ndbr 7974 . . . . . 6 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1412, 1, 13sylancr 587 . . . . 5 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
1511, 5, 14funcf1 17752 . . . 4 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
16 fcompt 7079 . . . 4 (( 1 :(Base‘𝐷)⟶V ∧ (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷)) → ( 1 ∘ (1st𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1st𝐹)‘𝑥))))
1710, 15, 16syl2anc 584 . . 3 (𝜑 → ( 1 ∘ (1st𝐹)) = (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1st𝐹)‘𝑥))))
18 eqid 2736 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
194adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
2015ffvelcdmda 7035 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
215, 18, 6, 19, 20catidcl 17562 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ( 1 ‘((1st𝐹)‘𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2221ralrimiva 3143 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)( 1 ‘((1st𝐹)‘𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
23 fvex 6855 . . . . 5 (Base‘𝐶) ∈ V
24 mptelixpg 8873 . . . . 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 233 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ( 1 ‘((1st𝐹)‘𝑥))) ∈ X𝑥 ∈ (Base‘𝐶)(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2717, 26eqeltrd 2838 . 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 1194 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑥 ∈ (Base‘𝐶))
3029, 20syldan 591 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
31 eqid 2736 . . . . . 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 1195 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
3432, 33ffvelcdmd 7036 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
35 eqid 2736 . . . . . . . 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 17754 . . . . . . 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 1196 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
3937, 38ffvelcdmd 7036 . . . . . 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 17563 . . . . 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 17564 . . . . 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 2779 . . . 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 6940 . . . . . 6 (((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑦 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐹))‘𝑦) = ( 1 ‘((1st𝐹)‘𝑦)))
4432, 33, 43syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ∘ (1st𝐹))‘𝑦) = ( 1 ‘((1st𝐹)‘𝑦)))
4544oveq1d 7372 . . . 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 6940 . . . . . 6 (((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐹))‘𝑥) = ( 1 ‘((1st𝐹)‘𝑥)))
4732, 29, 46syl2anc 584 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → (( 1 ∘ (1st𝐹))‘𝑥) = ( 1 ‘((1st𝐹)‘𝑥)))
4847oveq2d 7373 . . . 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 2786 . . 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 3200 . 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 17835 . 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 711 1 (𝜑 → ( 1 ∘ (1st𝐹)) ∈ (𝐹𝑁𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cop 4592   class class class wbr 5105  cmpt 5188  ccom 5637  Rel wrel 5638   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Xcixp 8835  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Idccid 17545   Func cfunc 17740   Nat cnat 17828   FuncCat cfuc 17829
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-ixp 8836  df-cat 17548  df-cid 17549  df-func 17744  df-nat 17830
This theorem is referenced by:  fuclid  17855  fucrid  17856  fuccatid  17858
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