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Theorem fuclid 17869
Description: Left identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fuclid.q 𝑄 = (𝐶 FuncCat 𝐷)
fuclid.n 𝑁 = (𝐶 Nat 𝐷)
fuclid.x = (comp‘𝑄)
fuclid.1 1 = (Id‘𝐷)
fuclid.r (𝜑𝑅 ∈ (𝐹𝑁𝐺))
Assertion
Ref Expression
fuclid (𝜑 → (( 1 ∘ (1st𝐺))(⟨𝐹, 𝐺 𝐺)𝑅) = 𝑅)

Proof of Theorem fuclid
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2731 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2731 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
3 relfunc 17762 . . . . . . . 8 Rel (𝐶 Func 𝐷)
4 fuclid.r . . . . . . . . . 10 (𝜑𝑅 ∈ (𝐹𝑁𝐺))
5 fuclid.n . . . . . . . . . . 11 𝑁 = (𝐶 Nat 𝐷)
65natrcl 17851 . . . . . . . . . 10 (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
74, 6syl 17 . . . . . . . . 9 (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
87simprd 496 . . . . . . . 8 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
9 1st2ndbr 7979 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
103, 8, 9sylancr 587 . . . . . . 7 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
111, 2, 10funcf1 17766 . . . . . 6 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
12 fvco3 6945 . . . . . 6 (((1st𝐺):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐺))‘𝑥) = ( 1 ‘((1st𝐺)‘𝑥)))
1311, 12sylan 580 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐺))‘𝑥) = ( 1 ‘((1st𝐺)‘𝑥)))
1413oveq1d 7377 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((( 1 ∘ (1st𝐺))‘𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(𝑅𝑥)) = (( 1 ‘((1st𝐺)‘𝑥))(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(𝑅𝑥)))
15 eqid 2731 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
16 fuclid.1 . . . . 5 1 = (Id‘𝐷)
177simpld 495 . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
18 funcrcl 17763 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1917, 18syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2019simprd 496 . . . . . 6 (𝜑𝐷 ∈ Cat)
2120adantr 481 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
22 1st2ndbr 7979 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
233, 17, 22sylancr 587 . . . . . . 7 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
241, 2, 23funcf1 17766 . . . . . 6 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
2524ffvelcdmda 7040 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
26 eqid 2731 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
2711ffvelcdmda 7040 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
285, 4nat1st2nd 17852 . . . . . . 7 (𝜑𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
2928adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
30 simpr 485 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
315, 29, 1, 15, 30natcl 17854 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑅𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
322, 15, 16, 21, 25, 26, 27, 31catlid 17577 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → (( 1 ‘((1st𝐺)‘𝑥))(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(𝑅𝑥)) = (𝑅𝑥))
3314, 32eqtrd 2771 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((( 1 ∘ (1st𝐺))‘𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(𝑅𝑥)) = (𝑅𝑥))
3433mpteq2dva 5210 . 2 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((( 1 ∘ (1st𝐺))‘𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(𝑅𝑥))) = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅𝑥)))
35 fuclid.q . . 3 𝑄 = (𝐶 FuncCat 𝐷)
36 fuclid.x . . 3 = (comp‘𝑄)
3735, 5, 16, 8fucidcl 17868 . . 3 (𝜑 → ( 1 ∘ (1st𝐺)) ∈ (𝐺𝑁𝐺))
3835, 5, 1, 26, 36, 4, 37fucco 17865 . 2 (𝜑 → (( 1 ∘ (1st𝐺))(⟨𝐹, 𝐺 𝐺)𝑅) = (𝑥 ∈ (Base‘𝐶) ↦ ((( 1 ∘ (1st𝐺))‘𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(𝑅𝑥))))
395, 28, 1natfn 17855 . . 3 (𝜑𝑅 Fn (Base‘𝐶))
40 dffn5 6906 . . 3 (𝑅 Fn (Base‘𝐶) ↔ 𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅𝑥)))
4139, 40sylib 217 . 2 (𝜑𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅𝑥)))
4234, 38, 413eqtr4d 2781 1 (𝜑 → (( 1 ∘ (1st𝐺))(⟨𝐹, 𝐺 𝐺)𝑅) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cop 4597   class class class wbr 5110  cmpt 5193  ccom 5642  Rel wrel 5643   Fn wfn 6496  wf 6497  cfv 6501  (class class class)co 7362  1st c1st 7924  2nd c2nd 7925  Basecbs 17094  Hom chom 17158  compcco 17159  Catccat 17558  Idccid 17559   Func cfunc 17754   Nat cnat 17842   FuncCat cfuc 17843
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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12423  df-z 12509  df-dec 12628  df-uz 12773  df-fz 13435  df-struct 17030  df-slot 17065  df-ndx 17077  df-base 17095  df-hom 17171  df-cco 17172  df-cat 17562  df-cid 17563  df-func 17758  df-nat 17844  df-fuc 17845
This theorem is referenced by:  fuccatid  17872
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