fuco2eld3 - Mathbox for Zhi Wang

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Theorem fuco2eld3 49210

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 49209	. . 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 2843	. 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 49207	. 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 1540 ∈ wcel 2109 ⟨cop 4603 class class class wbr 5115 × cxp 5644 Rel wrel 5651 ‘cfv 6519 1^st c1st 7975 2^nd c2nd 7976

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5259 ax-nul 5269 ax-pr 5395 ax-un 7718

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ral 3047 df-rex 3056 df-rab 3412 df-v 3457 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-nul 4305 df-if 4497 df-sn 4598 df-pr 4600 df-op 4604 df-uni 4880 df-br 5116 df-opab 5178 df-mpt 5197 df-id 5541 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-iota 6472 df-fun 6521 df-fv 6527 df-1st 7977 df-2nd 7978

This theorem is referenced by: fucof21 49242

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