fuco2eld3 - Mathbox for Zhi Wang

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Theorem fuco2eld3 50367

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
fuco2eld3	⊢ (𝜑 → ((1^st ‘(1^st ‘𝑈))𝑆(2^nd ‘(1^st ‘𝑈)) ∧ (1^st ‘(2^nd ‘𝑈))𝑅(2^nd ‘(2^nd ‘𝑈))))

**Proof of Theorem fuco2eld3**
Step	Hyp	Ref	Expression
1		fuco2eld2.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
2		fuco2eld.w	. . . 4 ⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅))
3		fuco2eld2.s	. . . 4 ⊢ Rel 𝑆
4		fuco2eld2.r	. . . 4 ⊢ Rel 𝑅
5	2, 1, 3, 4	fuco2eld2 50366	. . 3 ⊢ (𝜑 → 𝑈 = ⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩)
6	1, 5, 2	3eltr3d 2875	. 2 ⊢ (𝜑 → ⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩ ∈ (𝑆 × 𝑅))
7		fuco2el 50364	. 2 ⊢ (⟨⟨(1^st ‘(1^st ‘𝑈)), (2^nd ‘(1^st ‘𝑈))⟩, ⟨(1^st ‘(2^nd ‘𝑈)), (2^nd ‘(2^nd ‘𝑈))⟩⟩ ∈ (𝑆 × 𝑅) ↔ ((1^st ‘(1^st ‘𝑈))𝑆(2^nd ‘(1^st ‘𝑈)) ∧ (1^st ‘(2^nd ‘𝑈))𝑅(2^nd ‘(2^nd ‘𝑈))))
8	6, 7	sylib 221	1 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑈))𝑆(2^nd ‘(1^st ‘𝑈)) ∧ (1^st ‘(2^nd ‘𝑈))𝑅(2^nd ‘(2^nd ‘𝑈))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⟨cop 4590 class class class wbr 5103 × cxp 5649 Rel wrel 5656 ‘cfv 6531 1^st c1st 7988 2^nd c2nd 7989

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7740

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-iota 6487 df-fun 6533 df-fv 6539 df-1st 7990 df-2nd 7991

This theorem is used by: fucof21 50399

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