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Theorem fuclid 17985

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 2761	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2761	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		relfunc 17878	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
4		fuclid.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
5		fuclid.n	. . . . . . . . . . 11 ⊢ 𝑁 = (𝐶 Nat 𝐷)
6	5	natrcl 17969	. . . . . . . . . 10 ⊢ (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
7	4, 6	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
8	7	simprd 499	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
9		1st2ndbr 8019	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
10	3, 8, 9	sylancr 596	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
11	1, 2, 10	funcf1 17882	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
12		fvco3 6963	. . . . . 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 589	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1^st ‘𝐺))‘𝑥) = ( 1 ‘((1^st ‘𝐺)‘𝑥)))
14	13	oveq1d 7407	. . . 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 2761	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16		fuclid.1	. . . . 5 ⊢ 1 = (Id‘𝐷)
17	7	simpld 498	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
18		funcrcl 17879	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
20	19	simprd 499	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
21	20	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
22		1st2ndbr 8019	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
23	3, 17, 22	sylancr 596	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
24	1, 2, 23	funcf1 17882	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
25	24	ffvelcdmda 7061	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
26		eqid 2761	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
27	11	ffvelcdmda 7061	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
28	5, 4	nat1st2nd 17970	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
29	28	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
30		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
31	5, 29, 1, 15, 30	natcl 17972	. . . . 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 17698	. . . 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 2796	. . 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 5192	. 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 17984	. . 3 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐺)) ∈ (𝐺𝑁𝐺))
38	35, 5, 1, 26, 36, 4, 37	fucco 17981	. 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 17973	. . 3 ⊢ (𝜑 → 𝑅 Fn (Base‘𝐶))
40		dffn5 6921	. . 3 ⊢ (𝑅 Fn (Base‘𝐶) ↔ 𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅‘𝑥)))
41	39, 40	sylib 220	. 2 ⊢ (𝜑 → 𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅‘𝑥)))
42	34, 38, 41	3eqtr4d 2806	1 ⊢ (𝜑 → (( 1 ∘ (1^st ‘𝐺))(⟨𝐹, 𝐺⟩ ∙ 𝐺)𝑅) = 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ⟨cop 4587 class class class wbr 5099 ↦ cmpt 5180 ∘ ccom 5649 Rel wrel 5650 Fn wfn 6512 ⟶wf 6513 ‘cfv 6517 (class class class)co 7392 1^st c1st 7964 2^nd c2nd 7965 Basecbs 17228 Hom chom 17280 compcco 17281 Catccat 17679 Idccid 17680 Func cfunc 17870 Nat cnat 17960 FuncCat cfuc 17961

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8673 df-map 8805 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-z 12566 df-dec 12686 df-uz 12837 df-fz 13510 df-struct 17166 df-slot 17201 df-ndx 17213 df-base 17229 df-hom 17293 df-cco 17294 df-cat 17683 df-cid 17684 df-func 17874 df-nat 17962 df-fuc 17963

This theorem is referenced by: fuccatid 17988

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