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Theorem fuclid 18016

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	⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))

**Assertion**
Ref	Expression
fuclid	⊢ (𝜑 → (( 1 ∘ (1^st ‘𝐺))(⟨𝐹, 𝐺⟩ ∙ 𝐺)𝑅) = 𝑅)

Proof of Theorem fuclid Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2765	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2765	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		relfunc 17909	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
4		fuclid.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
5		fuclid.n	. . . . . . . . . . 11 ⊢ 𝑁 = (𝐶 Nat 𝐷)
6	5	natrcl 18000	. . . . . . . . . 10 ⊢ (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
7	4, 6	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
8	7	simprd 500	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
9		1st2ndbr 8027	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
10	3, 8, 9	sylancr 598	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
11	1, 2, 10	funcf1 17913	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
12		fvco3 6971	. . . . . 6 ⊢ (((1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1^st ‘𝐺))‘𝑥) = ( 1 ‘((1^st ‘𝐺)‘𝑥)))
13	11, 12	sylan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1^st ‘𝐺))‘𝑥) = ( 1 ‘((1^st ‘𝐺)‘𝑥)))
14	13	oveq1d 7415	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((( 1 ∘ (1^st ‘𝐺))‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑅‘𝑥)) = (( 1 ‘((1^st ‘𝐺)‘𝑥))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑅‘𝑥)))
15		eqid 2765	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16		fuclid.1	. . . . 5 ⊢ 1 = (Id‘𝐷)
17	7	simpld 499	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
18		funcrcl 17910	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
19	17, 18	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
20	19	simprd 500	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
21	20	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
22		1st2ndbr 8027	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
23	3, 17, 22	sylancr 598	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
24	1, 2, 23	funcf1 17913	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
25	24	ffvelcdmda 7069	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
26		eqid 2765	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
27	11	ffvelcdmda 7069	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
28	5, 4	nat1st2nd 18001	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
29	28	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
30		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
31	5, 29, 1, 15, 30	natcl 18003	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑅‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)))
32	2, 15, 16, 21, 25, 26, 27, 31	catlid 17729	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ‘((1^st ‘𝐺)‘𝑥))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑅‘𝑥)) = (𝑅‘𝑥))
33	14, 32	eqtrd 2800	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((( 1 ∘ (1^st ‘𝐺))‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑅‘𝑥)) = (𝑅‘𝑥))
34	33	mpteq2dva 5198	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((( 1 ∘ (1^st ‘𝐺))‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑅‘𝑥))) = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅‘𝑥)))
35		fuclid.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
36		fuclid.x	. . 3 ⊢ ∙ = (comp‘𝑄)
37	35, 5, 16, 8	fucidcl 18015	. . 3 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐺)) ∈ (𝐺𝑁𝐺))
38	35, 5, 1, 26, 36, 4, 37	fucco 18012	. 2 ⊢ (𝜑 → (( 1 ∘ (1^st ‘𝐺))(⟨𝐹, 𝐺⟩ ∙ 𝐺)𝑅) = (𝑥 ∈ (Base‘𝐶) ↦ ((( 1 ∘ (1^st ‘𝐺))‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑅‘𝑥))))
39	5, 28, 1	natfn 18004	. . 3 ⊢ (𝜑 → 𝑅 Fn (Base‘𝐶))
40		dffn5 6929	. . 3 ⊢ (𝑅 Fn (Base‘𝐶) ↔ 𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅‘𝑥)))
41	39, 40	sylib 221	. 2 ⊢ (𝜑 → 𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅‘𝑥)))
42	34, 38, 41	3eqtr4d 2810	1 ⊢ (𝜑 → (( 1 ∘ (1^st ‘𝐺))(⟨𝐹, 𝐺⟩ ∙ 𝐺)𝑅) = 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ⟨cop 4591 class class class wbr 5105 ↦ cmpt 5186 ∘ ccom 5656 Rel wrel 5657 Fn wfn 6520 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 1^st c1st 7972 2^nd c2nd 7973 Basecbs 17259 Hom chom 17311 compcco 17312 Catccat 17710 Idccid 17711 Func cfunc 17901 Nat cnat 17991 FuncCat cfuc 17992

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-er 8682 df-map 8814 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-fz 13527 df-struct 17197 df-slot 17232 df-ndx 17244 df-base 17260 df-hom 17324 df-cco 17325 df-cat 17714 df-cid 17715 df-func 17905 df-nat 17993 df-fuc 17994

This theorem is referenced by: fuccatid 18019

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