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Theorem fucrid 17674

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 2738	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2738	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		relfunc 17566	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
4		fuclid.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
5		fuclid.n	. . . . . . . . . . 11 ⊢ 𝑁 = (𝐶 Nat 𝐷)
6	5	natrcl 17655	. . . . . . . . . 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	simpld 495	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
9		1st2ndbr 7874	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
10	3, 8, 9	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
11	1, 2, 10	funcf1 17570	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
12		fvco3 6861	. . . . . 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 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1^st ‘𝐹))‘𝑥) = ( 1 ‘((1^st ‘𝐹)‘𝑥)))
14	13	oveq2d 7285	. . . 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 2738	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16		fuclid.1	. . . . 5 ⊢ 1 = (Id‘𝐷)
17		funcrcl 17567	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
18	8, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
19	18	simprd 496	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
20	19	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
21	11	ffvelrnda 6955	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
22		eqid 2738	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
23	7	simprd 496	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
24		1st2ndbr 7874	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
25	3, 23, 24	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
26	1, 2, 25	funcf1 17570	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
27	26	ffvelrnda 6955	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
28	5, 4	nat1st2nd 17656	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
29	28	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
30		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
31	5, 29, 1, 15, 30	natcl 17658	. . . . 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 17382	. . . 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 2778	. . 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 5175	. 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 17672	. . 3 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐹)) ∈ (𝐹𝑁𝐹))
38	35, 5, 1, 22, 36, 37, 4	fucco 17669	. 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 17659	. . 3 ⊢ (𝜑 → 𝑅 Fn (Base‘𝐶))
40		dffn5 6822	. . 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 → (𝑅(⟨𝐹, 𝐹⟩ ∙ 𝐺)( 1 ∘ (1^st ‘𝐹))) = 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ⟨cop 4569 class class class wbr 5075 ↦ cmpt 5158 ∘ ccom 5590 Rel wrel 5591 Fn wfn 6423 ⟶wf 6424 ‘cfv 6428 (class class class)co 7269 1^st c1st 7820 2^nd c2nd 7821 Basecbs 16901 Hom chom 16962 compcco 16963 Catccat 17362 Idccid 17363 Func cfunc 17558 Nat cnat 17646 FuncCat cfuc 17647

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8487 df-map 8606 df-ixp 8675 df-en 8723 df-dom 8724 df-sdom 8725 df-fin 8726 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-nn 11963 df-2 12025 df-3 12026 df-4 12027 df-5 12028 df-6 12029 df-7 12030 df-8 12031 df-9 12032 df-n0 12223 df-z 12309 df-dec 12427 df-uz 12572 df-fz 13229 df-struct 16837 df-slot 16872 df-ndx 16884 df-base 16902 df-hom 16975 df-cco 16976 df-cat 17366 df-cid 17367 df-func 17562 df-nat 17648 df-fuc 17649

This theorem is referenced by: fuccatid 17676

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