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Theorem fucrid 17674
Description: Right 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
fucrid (𝜑 → (𝑅(⟨𝐹, 𝐹 𝐺)( 1 ∘ (1st𝐹))) = 𝑅)

Proof of Theorem fucrid
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2738 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
3 relfunc 17566 . . . . . . . 8 Rel (𝐶 Func 𝐷)
4 fuclid.r . . . . . . . . . 10 (𝜑𝑅 ∈ (𝐹𝑁𝐺))
5 fuclid.n . . . . . . . . . . 11 𝑁 = (𝐶 Nat 𝐷)
65natrcl 17655 . . . . . . . . . 10 (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
74, 6syl 17 . . . . . . . . 9 (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
87simpld 495 . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
9 1st2ndbr 7874 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
103, 8, 9sylancr 587 . . . . . . 7 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
111, 2, 10funcf1 17570 . . . . . 6 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
12 fvco3 6861 . . . . . 6 (((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐹))‘𝑥) = ( 1 ‘((1st𝐹)‘𝑥)))
1311, 12sylan 580 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐹))‘𝑥) = ( 1 ‘((1st𝐹)‘𝑥)))
1413oveq2d 7285 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑅𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(( 1 ∘ (1st𝐹))‘𝑥)) = ((𝑅𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))( 1 ‘((1st𝐹)‘𝑥))))
15 eqid 2738 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
16 fuclid.1 . . . . 5 1 = (Id‘𝐷)
17 funcrcl 17567 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
188, 17syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1918simprd 496 . . . . . 6 (𝜑𝐷 ∈ Cat)
2019adantr 481 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
2111ffvelrnda 6955 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
22 eqid 2738 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
237simprd 496 . . . . . . . 8 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
24 1st2ndbr 7874 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
253, 23, 24sylancr 587 . . . . . . 7 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
261, 2, 25funcf1 17570 . . . . . 6 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
2726ffvelrnda 6955 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
285, 4nat1st2nd 17656 . . . . . . 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 17658 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑅𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
322, 15, 16, 20, 21, 22, 27, 31catrid 17382 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑅𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))( 1 ‘((1st𝐹)‘𝑥))) = (𝑅𝑥))
3314, 32eqtrd 2778 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑅𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(( 1 ∘ (1st𝐹))‘𝑥)) = (𝑅𝑥))
3433mpteq2dva 5175 . 2 (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((𝑅𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(( 1 ∘ (1st𝐹))‘𝑥))) = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅𝑥)))
35 fuclid.q . . 3 𝑄 = (𝐶 FuncCat 𝐷)
36 fuclid.x . . 3 = (comp‘𝑄)
3735, 5, 16, 8fucidcl 17672 . . 3 (𝜑 → ( 1 ∘ (1st𝐹)) ∈ (𝐹𝑁𝐹))
3835, 5, 1, 22, 36, 37, 4fucco 17669 . 2 (𝜑 → (𝑅(⟨𝐹, 𝐹 𝐺)( 1 ∘ (1st𝐹))) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑅𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(( 1 ∘ (1st𝐹))‘𝑥))))
395, 28, 1natfn 17659 . . 3 (𝜑𝑅 Fn (Base‘𝐶))
40 dffn5 6822 . . 3 (𝑅 Fn (Base‘𝐶) ↔ 𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅𝑥)))
4139, 40sylib 217 . 2 (𝜑𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅𝑥)))
4234, 38, 413eqtr4d 2788 1 (𝜑 → (𝑅(⟨𝐹, 𝐹 𝐺)( 1 ∘ (1st𝐹))) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  cop 4569   class class class wbr 5075  cmpt 5158  ccom 5590  Rel wrel 5591   Fn wfn 6423  wf 6424  cfv 6428  (class class class)co 7269  1st c1st 7820  2nd c2nd 7821  Basecbs 16901  Hom chom 16962  compcco 16963  Catccat 17362  Idccid 17363   Func cfunc 17558   Nat cnat 17646   FuncCat cfuc 17647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5210  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7580  ax-cnex 10916  ax-resscn 10917  ax-1cn 10918  ax-icn 10919  ax-addcl 10920  ax-addrcl 10921  ax-mulcl 10922  ax-mulrcl 10923  ax-mulcom 10924  ax-addass 10925  ax-mulass 10926  ax-distr 10927  ax-i2m1 10928  ax-1ne0 10929  ax-1rid 10930  ax-rnegex 10931  ax-rrecex 10932  ax-cnre 10933  ax-pre-lttri 10934  ax-pre-lttrn 10935  ax-pre-ltadd 10936  ax-pre-mulgt0 10937
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4842  df-iun 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5486  df-eprel 5492  df-po 5500  df-so 5501  df-fr 5541  df-we 5543  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-pred 6197  df-ord 6264  df-on 6265  df-lim 6266  df-suc 6267  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-f1 6433  df-fo 6434  df-f1o 6435  df-fv 6436  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7705  df-1st 7822  df-2nd 7823  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-1o 8286  df-er 8487  df-map 8606  df-ixp 8675  df-en 8723  df-dom 8724  df-sdom 8725  df-fin 8726  df-pnf 11000  df-mnf 11001  df-xr 11002  df-ltxr 11003  df-le 11004  df-sub 11196  df-neg 11197  df-nn 11963  df-2 12025  df-3 12026  df-4 12027  df-5 12028  df-6 12029  df-7 12030  df-8 12031  df-9 12032  df-n0 12223  df-z 12309  df-dec 12427  df-uz 12572  df-fz 13229  df-struct 16837  df-slot 16872  df-ndx 16884  df-base 16902  df-hom 16975  df-cco 16976  df-cat 17366  df-cid 17367  df-func 17562  df-nat 17648  df-fuc 17649
This theorem is referenced by:  fuccatid  17676
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