fuco2eld2 - Mathbox for Zhi Wang

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Theorem fuco2eld2 49662

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 2839	. . 3 ⊢ (𝜑 → 𝑈 ∈ (𝑆 × 𝑅))
4		1st2nd2 7982	. . 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 5639	. . . . . 6 ⊢ (Rel 𝑆 ↔ 𝑆 ⊆ (V × V))
8	6, 7	mpbi 230	. . . . 5 ⊢ 𝑆 ⊆ (V × V)
9		xp1st 7975	. . . . 5 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (1^st ‘𝑈) ∈ 𝑆)
10	8, 9	sselid 3933	. . . 4 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (1^st ‘𝑈) ∈ (V × V))
11		1st2nd2 7982	. . . 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 5639	. . . . . 6 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
15	13, 14	mpbi 230	. . . . 5 ⊢ 𝑅 ⊆ (V × V)
16		xp2nd 7976	. . . . 5 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (2^nd ‘𝑈) ∈ 𝑅)
17	15, 16	sselid 3933	. . . 4 ⊢ (𝑈 ∈ (𝑆 × 𝑅) → (2^nd ‘𝑈) ∈ (V × V))
18		1st2nd2 7982	. . . 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 4839	. 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 2772	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 1542 ∈ wcel 2114 Vcvv 3442 ⊆ wss 3903 ⟨cop 4588 × cxp 5630 Rel wrel 5637 ‘cfv 6500 1^st c1st 7941 2^nd c2nd 7942

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-iota 6456 df-fun 6502 df-fv 6508 df-1st 7943 df-2nd 7944

This theorem is referenced by: fuco2eld3 49663 fucof21 49695 fucoid2 49697

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