isnat2 - Metamath Proof Explorer

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Theorem isnat2 17664

Description: Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.)

**Hypotheses**
Ref	Expression
natfval.1	⊢ 𝑁 = (𝐶 Nat 𝐷)
natfval.b	⊢ 𝐵 = (Base‘𝐶)
natfval.h	⊢ 𝐻 = (Hom ‘𝐶)
natfval.j	⊢ 𝐽 = (Hom ‘𝐷)
natfval.o	⊢ · = (comp‘𝐷)
isnat2.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
isnat2.g	⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))

**Assertion**
Ref	Expression
isnat2	⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥)))))

Distinct variable groups: 𝑥,ℎ,𝑦,𝐴 𝑥,𝐵,𝑦 𝐶,ℎ,𝑥,𝑦 ℎ,𝐹,𝑥,𝑦 ℎ,𝐺,𝑥,𝑦 ℎ,𝐻 𝜑,ℎ,𝑥,𝑦 𝐷,ℎ,𝑥,𝑦 Allowed substitution hints: 𝐵(ℎ) · (𝑥,𝑦,ℎ) 𝐻(𝑥,𝑦) 𝐽(𝑥,𝑦,ℎ) 𝑁(𝑥,𝑦,ℎ)

**Proof of Theorem isnat2**
Step	Hyp	Ref	Expression
1		relfunc 17577	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
2		isnat2.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
3		1st2nd 7880	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	1, 2, 3	sylancr 587	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
5		isnat2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
6		1st2nd 7880	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
7	1, 5, 6	sylancr 587	. . . 4 ⊢ (𝜑 → 𝐺 = ⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)
8	4, 7	oveq12d 7293	. . 3 ⊢ (𝜑 → (𝐹𝑁𝐺) = (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
9	8	eleq2d 2824	. 2 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ 𝐴 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩)))
10		natfval.1	. . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷)
11		natfval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
12		natfval.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
13		natfval.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐷)
14		natfval.o	. . 3 ⊢ · = (comp‘𝐷)
15		1st2ndbr 7883	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
16	1, 2, 15	sylancr 587	. . 3 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
17		1st2ndbr 7883	. . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
18	1, 5, 17	sylancr 587	. . 3 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
19	10, 11, 12, 13, 14, 16, 18	isnat 17663	. 2 ⊢ (𝜑 → (𝐴 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥)))))
20	9, 19	bitrd 278	1 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 (((1^st ‘𝐹)‘𝑥)𝐽((1^st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐹)‘𝑦)⟩ · ((1^st ‘𝐺)‘𝑦))((𝑥(2^nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2^nd ‘𝐺)𝑦)‘ℎ)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩ · ((1^st ‘𝐺)‘𝑦))(𝐴‘𝑥)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ⟨cop 4567 class class class wbr 5074 Rel wrel 5594 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 Xcixp 8685 Basecbs 16912 Hom chom 16973 compcco 16974 Func cfunc 17569 Nat cnat 17657

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 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 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-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-ixp 8686 df-func 17573 df-nat 17659

This theorem is referenced by: fuccocl 17682 fucidcl 17683 invfuc 17692 curf2cl 17949 yonedalem4c 17995 yonedalem3 17998

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