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Theorem fucrid 16979
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 2825 . . . . . . 7 (Base‘𝐶) = (Base‘𝐶)
2 eqid 2825 . . . . . . 7 (Base‘𝐷) = (Base‘𝐷)
3 relfunc 16874 . . . . . . . 8 Rel (𝐶 Func 𝐷)
4 fuclid.r . . . . . . . . . 10 (𝜑𝑅 ∈ (𝐹𝑁𝐺))
5 fuclid.n . . . . . . . . . . 11 𝑁 = (𝐶 Nat 𝐷)
65natrcl 16962 . . . . . . . . . 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 490 . . . . . . . 8 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
9 1st2ndbr 7479 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
103, 8, 9sylancr 583 . . . . . . 7 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
111, 2, 10funcf1 16878 . . . . . 6 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
12 fvco3 6522 . . . . . 6 (((1st𝐹):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐹))‘𝑥) = ( 1 ‘((1st𝐹)‘𝑥)))
1311, 12sylan 577 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1st𝐹))‘𝑥) = ( 1 ‘((1st𝐹)‘𝑥)))
1413oveq2d 6921 . . . 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 2825 . . . . 5 (Hom ‘𝐷) = (Hom ‘𝐷)
16 fuclid.1 . . . . 5 1 = (Id‘𝐷)
17 funcrcl 16875 . . . . . . . 8 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
188, 17syl 17 . . . . . . 7 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
1918simprd 491 . . . . . 6 (𝜑𝐷 ∈ Cat)
2019adantr 474 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
2111ffvelrnda 6608 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
22 eqid 2825 . . . . 5 (comp‘𝐷) = (comp‘𝐷)
237simprd 491 . . . . . . . 8 (𝜑𝐺 ∈ (𝐶 Func 𝐷))
24 1st2ndbr 7479 . . . . . . . 8 ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
253, 23, 24sylancr 583 . . . . . . 7 (𝜑 → (1st𝐺)(𝐶 Func 𝐷)(2nd𝐺))
261, 2, 25funcf1 16878 . . . . . 6 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐷))
2726ffvelrnda 6608 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐷))
285, 4nat1st2nd 16963 . . . . . . 7 (𝜑𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
2928adantr 474 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1st𝐹), (2nd𝐹)⟩𝑁⟨(1st𝐺), (2nd𝐺)⟩))
30 simpr 479 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
315, 29, 1, 15, 30natcl 16965 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → (𝑅𝑥) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐺)‘𝑥)))
322, 15, 16, 20, 21, 22, 27, 31catrid 16697 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑅𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))( 1 ‘((1st𝐹)‘𝑥))) = (𝑅𝑥))
3314, 32eqtrd 2861 . . 3 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑅𝑥)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑥)⟩(comp‘𝐷)((1st𝐺)‘𝑥))(( 1 ∘ (1st𝐹))‘𝑥)) = (𝑅𝑥))
3433mpteq2dva 4967 . 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 16977 . . 3 (𝜑 → ( 1 ∘ (1st𝐹)) ∈ (𝐹𝑁𝐹))
3835, 5, 1, 22, 36, 37, 4fucco 16974 . 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 16966 . . 3 (𝜑𝑅 Fn (Base‘𝐶))
40 dffn5 6488 . . 3 (𝑅 Fn (Base‘𝐶) ↔ 𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅𝑥)))
4139, 40sylib 210 . 2 (𝜑𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅𝑥)))
4234, 38, 413eqtr4d 2871 1 (𝜑 → (𝑅(⟨𝐹, 𝐹 𝐺)( 1 ∘ (1st𝐹))) = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  cop 4403   class class class wbr 4873  cmpt 4952  ccom 5346  Rel wrel 5347   Fn wfn 6118  wf 6119  cfv 6123  (class class class)co 6905  1st c1st 7426  2nd c2nd 7427  Basecbs 16222  Hom chom 16316  compcco 16317  Catccat 16677  Idccid 16678   Func cfunc 16866   Nat cnat 16953   FuncCat cfuc 16954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-map 8124  df-ixp 8176  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-z 11705  df-dec 11822  df-uz 11969  df-fz 12620  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-hom 16329  df-cco 16330  df-cat 16681  df-cid 16682  df-func 16870  df-nat 16955  df-fuc 16956
This theorem is referenced by:  fuccatid  16981
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