fuco2eld2 - Mathbox for Zhi Wang

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Theorem fuco2eld2 49943

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 2867	. . 3 ⊢ (𝜑 → 𝑈 ∈ (𝑆 × 𝑅))
4		1st2nd2 8013	. . 3 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
5	3, 4	syl 18	. 2 ⊢ (𝜑 → 𝑈 = ⟨(1^st ‘𝑈), (2^nd ‘𝑈)⟩)
6		fuco2eld2.s	. . . . . 6 ⊢ Rel 𝑆
7		df-rel 5659	. . . . . 6 ⊢ (Rel 𝑆 ↔ 𝑆 ⊆ (V × V))
8	6, 7	mpbi 233	. . . . 5 ⊢ 𝑆 ⊆ (V × V)
9		xp1st 8006	. . . . 5 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (1^st ‘𝑈) ∈ 𝑆)
10	8, 9	sselid 3937	. . . 4 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (1^st ‘𝑈) ∈ (V × V))
11		1st2nd2 8013	. . . 4 ⊢ ((1^st ‘𝑈) ∈ (V × V) → (1^st ‘𝑈) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩)
12	3, 10, 11	3syl 19	. . 3 ⊢ (𝜑 → (1^st ‘𝑈) = ⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩)
13		fuco2eld2.r	. . . . . 6 ⊢ Rel 𝑅
14		df-rel 5659	. . . . . 6 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
15	13, 14	mpbi 233	. . . . 5 ⊢ 𝑅 ⊆ (V × V)
16		xp2nd 8007	. . . . 5 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (2^nd ‘𝑈) ∈ 𝑅)
17	15, 16	sselid 3937	. . . 4 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (2^nd ‘𝑈) ∈ (V × V))
18		1st2nd2 8013	. . . 4 ⊢ ((2^nd ‘𝑈) ∈ (V × V) → (2^nd ‘𝑈) = ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩)
19	3, 17, 18	3syl 19	. . 3 ⊢ (𝜑 → (2^nd ‘𝑈) = ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩)
20	12, 19	opeq12d 4842	. 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 2800	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 1563 ∈ wcel 2145 Vcvv 3457 ⊆ wss 3907 ⟨cop 4591 × cxp 5650 Rel wrel 5657 ‘cfv 6525 1^st c1st 7972 2^nd c2nd 7973

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-nul 5261 ax-pr 5395 ax-un 7722

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-br 5106 df-opab 5168 df-mpt 5187 df-id 5547 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-iota 6481 df-fun 6527 df-fv 6533 df-1st 7974 df-2nd 7975

This theorem is referenced by: fuco2eld3 49944 fucof21 49976 fucoid2 49978

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