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Theorem fucrid 18028

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	⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))

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

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

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		relfunc 17920	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
4		fuclid.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
5		fuclid.n	. . . . . . . . . . 11 ⊢ 𝑁 = (𝐶 Nat 𝐷)
6	5	natrcl 18011	. . . . . . . . . 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	simpld 499	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
9		1st2ndbr 8040	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
10	3, 8, 9	sylancr 598	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
11	1, 2, 10	funcf1 17924	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
12		fvco3 6983	. . . . . 6 ⊢ (((1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1^st ‘𝐹))‘𝑥) = ( 1 ‘((1^st ‘𝐹)‘𝑥)))
13	11, 12	sylan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1^st ‘𝐹))‘𝑥) = ( 1 ‘((1^st ‘𝐹)‘𝑥)))
14	13	oveq2d 7428	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑅‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(( 1 ∘ (1^st ‘𝐹))‘𝑥)) = ((𝑅‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))( 1 ‘((1^st ‘𝐹)‘𝑥))))
15		eqid 2763	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16		fuclid.1	. . . . 5 ⊢ 1 = (Id‘𝐷)
17		funcrcl 17921	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
18	8, 17	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
19	18	simprd 500	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
20	19	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
21	11	ffvelcdmda 7081	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
22		eqid 2763	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
23	7	simprd 500	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
24		1st2ndbr 8040	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
25	3, 23, 24	sylancr 598	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
26	1, 2, 25	funcf1 17924	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
27	26	ffvelcdmda 7081	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
28	5, 4	nat1st2nd 18012	. . . . . . 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 18014	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑅‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)))
32	2, 15, 16, 20, 21, 22, 27, 31	catrid 17741	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑅‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))( 1 ‘((1^st ‘𝐹)‘𝑥))) = (𝑅‘𝑥))
33	14, 32	eqtrd 2798	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑅‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(( 1 ∘ (1^st ‘𝐹))‘𝑥)) = (𝑅‘𝑥))
34	33	mpteq2dva 5205	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((𝑅‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(( 1 ∘ (1^st ‘𝐹))‘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 18026	. . 3 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) ∈ (𝐹𝑁𝐹))
38	35, 5, 1, 22, 36, 37, 4	fucco 18023	. 2 ⊢ (𝜑 → (𝑅(⟨𝐹, 𝐹⟩ ∙ 𝐺)( 1 ∘ (1^st ‘𝐹))) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝑅‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(( 1 ∘ (1^st ‘𝐹))‘𝑥))))
39	5, 28, 1	natfn 18015	. . 3 ⊢ (𝜑 → 𝑅 Fn (Base‘𝐶))
40		dffn5 6941	. . 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 2808	1 ⊢ (𝜑 → (𝑅(⟨𝐹, 𝐹⟩ ∙ 𝐺)( 1 ∘ (1^st ‘𝐹))) = 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ⟨cop 4596 class class class wbr 5110 ↦ cmpt 5193 ∘ ccom 5667 Rel wrel 5668 Fn wfn 6533 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 1^st c1st 7985 2^nd c2nd 7986 Basecbs 17270 Hom chom 17322 compcco 17323 Catccat 17721 Idccid 17722 Func cfunc 17912 Nat cnat 18002 FuncCat cfuc 18003

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-map 8827 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-fz 13537 df-struct 17208 df-slot 17243 df-ndx 17255 df-base 17271 df-hom 17335 df-cco 17336 df-cat 17725 df-cid 17726 df-func 17916 df-nat 18004 df-fuc 18005

This theorem is referenced by: fuccatid 18030

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