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Theorem fucidcl 17757
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 17652 . . . . . . . 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 17462 . . . . . 6 (𝐷 ∈ Cat → 1 Fn (Base‘𝐷))
84, 7syl 17 . . . . 5 (𝜑1 Fn (Base‘𝐷))
9 dffn2 6639 . . . . 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 17651 . . . . . 6 Rel (𝐶 Func 𝐷)
13 1st2ndbr 7929 . . . . . 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 17655 . . . 4 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
16 fcompt 7044 . . . 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 7000 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
215, 18, 6, 19, 20catidcl 17465 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ( 1 ‘((1st𝐹)‘𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
2221ralrimiva 3139 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)( 1 ‘((1st𝐹)‘𝑥)) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑥)))
23 fvex 6824 . . . . 5 (Base‘𝐶) ∈ V
24 mptelixpg 8772 . . . . 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 2837 . 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 1193 . . . . . . 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 1194 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑦 ∈ (Base‘𝐶))
3432, 33ffvelcdmd 7001 . . . . . 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 17657 . . . . . . 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 1195 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
3937, 38ffvelcdmd 7001 . . . . . 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 17466 . . . . 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 17467 . . . . 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 6906 . . . . . 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 7331 . . . 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 6906 . . . . . 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 7332 . . . 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 3196 . 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 17738 . 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 710 1 (𝜑 → ( 1 ∘ (1st𝐹)) ∈ (𝐹𝑁𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wral 3061  Vcvv 3440  cop 4576   class class class wbr 5086  cmpt 5169  ccom 5611  Rel wrel 5612   Fn wfn 6460  wf 6461  cfv 6465  (class class class)co 7316  1st c1st 7875  2nd c2nd 7876  Xcixp 8734  Basecbs 16986  Hom chom 17047  compcco 17048  Catccat 17447  Idccid 17448   Func cfunc 17643   Nat cnat 17731   FuncCat cfuc 17732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5170  df-id 5506  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-riota 7273  df-ov 7319  df-oprab 7320  df-mpo 7321  df-1st 7877  df-2nd 7878  df-map 8666  df-ixp 8735  df-cat 17451  df-cid 17452  df-func 17647  df-nat 17733
This theorem is referenced by:  fuclid  17758  fucrid  17759  fuccatid  17761
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