fuco2eld3 - Mathbox for Zhi Wang

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Theorem fuco2eld3 48970

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 48969	. . 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 2847	. 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 48967	. 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 218	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

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⟨cop 4614 class class class wbr 5125 × cxp 5665 Rel wrel 5672 ‘cfv 6542 1^st c1st 7995 2^nd c2nd 7996

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5278 ax-nul 5288 ax-pr 5414 ax-un 7738

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ral 3051 df-rex 3060 df-rab 3421 df-v 3466 df-dif 3936 df-un 3938 df-in 3940 df-ss 3950 df-nul 4316 df-if 4508 df-sn 4609 df-pr 4611 df-op 4615 df-uni 4890 df-br 5126 df-opab 5188 df-mpt 5208 df-id 5560 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-iota 6495 df-fun 6544 df-fv 6550 df-1st 7997 df-2nd 7998

This theorem is referenced by: fucof21 49002

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