fuco2eld2 - Mathbox for Zhi Wang

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Theorem fuco2eld2 48883

Description: Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.)

**Hypotheses**
Ref	Expression
fuco2eld.w	⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅))
fuco2eld2.u	⊢ (𝜑 → 𝑈 ∈ 𝑊)
fuco2eld2.s	⊢ Rel 𝑆
fuco2eld2.r	⊢ Rel 𝑅

**Assertion**
Ref	Expression
fuco2eld2	⊢ (𝜑 → 𝑈 = ⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩)

**Proof of Theorem fuco2eld2**
Step	Hyp	Ref	Expression
1		fuco2eld2.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
2		fuco2eld.w	. . . 4 ⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅))
3	1, 2	eleqtrd 2843	. . 3 ⊢ (𝜑 → 𝑈 ∈ (𝑆 × 𝑅))
4		1st2nd2 8061	. . 3 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
5	3, 4	syl 17	. 2 ⊢ (𝜑 → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
6		fuco2eld2.s	. . . . . 6 ⊢ Rel 𝑆
7		df-rel 5700	. . . . . 6 ⊢ (Rel 𝑆 ↔ 𝑆 ⊆ (V × V))
8	6, 7	mpbi 230	. . . . 5 ⊢ 𝑆 ⊆ (V × V)
9		xp1st 8054	. . . . 5 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (1^st ‘𝑈) ∈ 𝑆)
10	8, 9	sselid 3996	. . . 4 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (1^st ‘𝑈) ∈ (V × V))
11		1st2nd2 8061	. . . 4 ⊢ ((1^st ‘𝑈) ∈ (V × V) → (1^st ‘𝑈) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩)
12	3, 10, 11	3syl 18	. . 3 ⊢ (𝜑 → (1^st ‘𝑈) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩)
13		fuco2eld2.r	. . . . . 6 ⊢ Rel 𝑅
14		df-rel 5700	. . . . . 6 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
15	13, 14	mpbi 230	. . . . 5 ⊢ 𝑅 ⊆ (V × V)
16		xp2nd 8055	. . . . 5 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (2^nd ‘𝑈) ∈ 𝑅)
17	15, 16	sselid 3996	. . . 4 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (2^nd ‘𝑈) ∈ (V × V))
18		1st2nd2 8061	. . . 4 ⊢ ((2^nd ‘𝑈) ∈ (V × V) → (2^nd ‘𝑈) = ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩)
19	3, 17, 18	3syl 18	. . 3 ⊢ (𝜑 → (2^nd ‘𝑈) = ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩)
20	12, 19	opeq12d 4889	. 2 ⊢ (𝜑 → ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩ = ⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩)
21	5, 20	eqtrd 2777	1 ⊢ (𝜑 → 𝑈 = ⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 Vcvv 3481 ⊆ wss 3966 ⟨cop 4640 × cxp 5691 Rel wrel 5698 ‘cfv 6569 1^st c1st 8020 2^nd c2nd 8021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5305 ax-nul 5315 ax-pr 5441 ax-un 7761

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3483 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-nul 4343 df-if 4535 df-sn 4635 df-pr 4637 df-op 4641 df-uni 4916 df-br 5152 df-opab 5214 df-mpt 5235 df-id 5587 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-iota 6522 df-fun 6571 df-fv 6577 df-1st 8022 df-2nd 8023

This theorem is referenced by: fuco2eld3 48884 fucof21 48914 fucoid2 48916

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