natcoeq - Mathbox for Zhi Wang

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Theorem natcoeq 50437

Description: A natural transformation equals a composite exactly when all its components equal the corresponding composites, including when the object set is empty. (Contributed by Zhi Wang, 14-Sep-2026.)

**Hypotheses**
Ref	Expression
natideq.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
natideq.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
natideq.b	⊢ 𝐵 = (Base‘𝐶)
natcoeq.o	⊢ ⚬ = (comp‘𝐷)
natcoeq.x	⊢ · = (comp‘𝑄)
natcoeq.a	⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
natcoeq.r	⊢ (𝜑 → 𝑅 ∈ (𝐺𝑁𝐻))
natcoeq.l	⊢ (𝜑 → 𝐿 ∈ (𝐹𝑁𝐻))

**Assertion**
Ref	Expression
natcoeq	⊢ (𝜑 → (𝐿 = (𝑅(⟨𝐹, 𝐺⟩ · 𝐻)𝐴) ↔ ∀𝑥 ∈ 𝐵 (𝐿‘𝑥) = ((𝑅‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ ⚬ ((1^st ‘𝐻)‘𝑥))(𝐴‘𝑥))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝑥,𝐷 𝑥,𝐹 𝑥,𝑄 𝑥,𝑁 𝜑,𝑥 𝑥,𝐺 𝑥,𝐻 𝑥,𝐿 𝑥,𝑅 𝑥, ⚬ 𝑥, ·

**Proof of Theorem natcoeq**
Step	Hyp	Ref	Expression
1		natideq.n	. . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷)
2		natideq.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		natcoeq.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝐹𝑁𝐻))
4		natideq.q	. . . 4 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
5		natcoeq.x	. . . 4 ⊢ · = (comp‘𝑄)
6		natcoeq.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐹𝑁𝐺))
7		natcoeq.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ (𝐺𝑁𝐻))
8	4, 1, 5, 6, 7	fuccocl 18144	. . 3 ⊢ (𝜑 → (𝑅(⟨𝐹, 𝐺⟩ · 𝐻)𝐴) ∈ (𝐹𝑁𝐻))
9	1, 2, 3, 8	nateq 50435	. 2 ⊢ (𝜑 → (𝐿 = (𝑅(⟨𝐹, 𝐺⟩ · 𝐻)𝐴) ↔ ∀𝑥 ∈ 𝐵 (𝐿‘𝑥) = ((𝑅(⟨𝐹, 𝐺⟩ · 𝐻)𝐴)‘𝑥)))
10		natcoeq.o	. . . . 5 ⊢ ⚬ = (comp‘𝐷)
11	6	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ (𝐹𝑁𝐺))
12	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ (𝐺𝑁𝐻))
13		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
14	4, 1, 2, 10, 5, 11, 12, 13	fuccoval 18143	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝑅(⟨𝐹, 𝐺⟩ · 𝐻)𝐴)‘𝑥) = ((𝑅‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ ⚬ ((1^st ‘𝐻)‘𝑥))(𝐴‘𝑥)))
15	14	eqeq2d 2771	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐿‘𝑥) = ((𝑅(⟨𝐹, 𝐺⟩ · 𝐻)𝐴)‘𝑥) ↔ (𝐿‘𝑥) = ((𝑅‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ ⚬ ((1^st ‘𝐻)‘𝑥))(𝐴‘𝑥))))
16	15	ralbidva 3183	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝐿‘𝑥) = ((𝑅(⟨𝐹, 𝐺⟩ · 𝐻)𝐴)‘𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝐿‘𝑥) = ((𝑅‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ ⚬ ((1^st ‘𝐻)‘𝑥))(𝐴‘𝑥))))
17	9, 16	bitrd 282	1 ⊢ (𝜑 → (𝐿 = (𝑅(⟨𝐹, 𝐺⟩ · 𝐻)𝐴) ↔ ∀𝑥 ∈ 𝐵 (𝐿‘𝑥) = ((𝑅‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ ⚬ ((1^st ‘𝐻)‘𝑥))(𝐴‘𝑥))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ⟨cop 4589 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 Basecbs 17389 compcco 17442 Nat cnat 18121 FuncCat cfuc 18122

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11258 ax-resscn 11259 ax-1cn 11260 ax-icn 11261 ax-addcl 11262 ax-addrcl 11263 ax-mulcl 11264 ax-mulrcl 11265 ax-mulcom 11266 ax-addass 11267 ax-mulass 11268 ax-distr 11269 ax-i2m1 11270 ax-1ne0 11271 ax-1rid 11272 ax-rnegex 11273 ax-rrecex 11274 ax-cnre 11275 ax-pre-lttri 11276 ax-pre-lttrn 11277 ax-pre-ltadd 11278 ax-pre-mulgt0 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 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 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8301 df-wrecs 8332 df-recs 8381 df-rdg 8420 df-1o 8478 df-er 8719 df-map 8851 df-ixp 8928 df-en 8976 df-dom 8977 df-sdom 8978 df-fin 8979 df-pnf 11347 df-mnf 11348 df-xr 11349 df-ltxr 11350 df-le 11351 df-sub 11545 df-neg 11546 df-nn 12338 df-2 12407 df-3 12408 df-4 12409 df-5 12410 df-6 12411 df-7 12412 df-8 12413 df-9 12414 df-n0 12609 df-z 12696 df-dec 12817 df-uz 12968 df-fz 13642 df-struct 17327 df-slot 17362 df-ndx 17374 df-base 17390 df-hom 17454 df-cco 17455 df-cat 17844 df-func 18035 df-nat 18123 df-fuc 18124

This theorem is used by: fucidcobi 50438

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