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Theorem fuclid 17600

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 2738	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2738	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		relfunc 17493	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
4		fuclid.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
5		fuclid.n	. . . . . . . . . . 11 ⊢ 𝑁 = (𝐶 Nat 𝐷)
6	5	natrcl 17582	. . . . . . . . . 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 495	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
9		1st2ndbr 7856	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
10	3, 8, 9	sylancr 586	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
11	1, 2, 10	funcf1 17497	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
12		fvco3 6849	. . . . . 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 579	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1^st ‘𝐺))‘𝑥) = ( 1 ‘((1^st ‘𝐺)‘𝑥)))
14	13	oveq1d 7270	. . . 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 2738	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16		fuclid.1	. . . . 5 ⊢ 1 = (Id‘𝐷)
17	7	simpld 494	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
18		funcrcl 17494	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
20	19	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
21	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
22		1st2ndbr 7856	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
23	3, 17, 22	sylancr 586	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
24	1, 2, 23	funcf1 17497	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
25	24	ffvelrnda 6943	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
26		eqid 2738	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
27	11	ffvelrnda 6943	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
28	5, 4	nat1st2nd 17583	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
29	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
30		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
31	5, 29, 1, 15, 30	natcl 17585	. . . . 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 17309	. . . 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 2778	. . 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 5170	. 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 17599	. . 3 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐺)) ∈ (𝐺𝑁𝐺))
38	35, 5, 1, 26, 36, 4, 37	fucco 17596	. 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 17586	. . 3 ⊢ (𝜑 → 𝑅 Fn (Base‘𝐶))
40		dffn5 6810	. . 3 ⊢ (𝑅 Fn (Base‘𝐶) ↔ 𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅‘𝑥)))
41	39, 40	sylib 217	. 2 ⊢ (𝜑 → 𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅‘𝑥)))
42	34, 38, 41	3eqtr4d 2788	1 ⊢ (𝜑 → (( 1 ∘ (1^st ‘𝐺))(⟨𝐹, 𝐺⟩ ∙ 𝐺)𝑅) = 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ⟨cop 4564 class class class wbr 5070 ↦ cmpt 5153 ∘ ccom 5584 Rel wrel 5585 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 Basecbs 16840 Hom chom 16899 compcco 16900 Catccat 17290 Idccid 17291 Func cfunc 17485 Nat cnat 17573 FuncCat cfuc 17574

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-hom 16912 df-cco 16913 df-cat 17294 df-cid 17295 df-func 17489 df-nat 17575 df-fuc 17576

This theorem is referenced by: fuccatid 17603

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