fuco2eld2 - Mathbox for Zhi Wang

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Theorem fuco2eld2 49899

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 2863	. . 3 ⊢ (𝜑 → 𝑈 ∈ (𝑆 × 𝑅))
4		1st2nd2 8005	. . 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 5652	. . . . . 6 ⊢ (Rel 𝑆 ↔ 𝑆 ⊆ (V × V))
8	6, 7	mpbi 232	. . . . 5 ⊢ 𝑆 ⊆ (V × V)
9		xp1st 7998	. . . . 5 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (1^st ‘𝑈) ∈ 𝑆)
10	8, 9	sselid 3934	. . . 4 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (1^st ‘𝑈) ∈ (V × V))
11		1st2nd2 8005	. . . 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 5652	. . . . . 6 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
15	13, 14	mpbi 232	. . . . 5 ⊢ 𝑅 ⊆ (V × V)
16		xp2nd 7999	. . . . 5 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (2^nd ‘𝑈) ∈ 𝑅)
17	15, 16	sselid 3934	. . . 4 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (2^nd ‘𝑈) ∈ (V × V))
18		1st2nd2 8005	. . . 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 4838	. 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 2796	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 1559 ∈ wcel 2141 Vcvv 3453 ⊆ wss 3904 ⟨cop 4587 × cxp 5643 Rel wrel 5650 ‘cfv 6517 1^st c1st 7964 2^nd c2nd 7965

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-iota 6473 df-fun 6519 df-fv 6525 df-1st 7966 df-2nd 7967

This theorem is referenced by: fuco2eld3 49900 fucof21 49932 fucoid2 49934

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