fuco2eld2 - Mathbox for Zhi Wang

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Theorem fuco2eld2 49811

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 2842	. . 3 ⊢ (𝜑 → 𝑈 ∈ (𝑆 × 𝑅))
4		1st2nd2 7977	. . 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 5632	. . . . . 6 ⊢ (Rel 𝑆 ↔ 𝑆 ⊆ (V × V))
8	6, 7	mpbi 231	. . . . 5 ⊢ 𝑆 ⊆ (V × V)
9		xp1st 7970	. . . . 5 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (1^st ‘𝑈) ∈ 𝑆)
10	8, 9	sselid 3920	. . . 4 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (1^st ‘𝑈) ∈ (V × V))
11		1st2nd2 7977	. . . 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 5632	. . . . . 6 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
15	13, 14	mpbi 231	. . . . 5 ⊢ 𝑅 ⊆ (V × V)
16		xp2nd 7971	. . . . 5 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (2^nd ‘𝑈) ∈ 𝑅)
17	15, 16	sselid 3920	. . . 4 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (2^nd ‘𝑈) ∈ (V × V))
18		1st2nd2 7977	. . . 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 4819	. 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 2775	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 1547 ∈ wcel 2119 Vcvv 3432 ⊆ wss 3890 ⟨cop 4568 × cxp 5623 Rel wrel 5630 ‘cfv 6492 1^st c1st 7936 2^nd c2nd 7937

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pr 5369 ax-un 7685

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-iota 6448 df-fun 6494 df-fv 6500 df-1st 7938 df-2nd 7939

This theorem is referenced by: fuco2eld3 49812 fucof21 49844 fucoid2 49846

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