Theorem fuco2eld 49898

Description: Equivalence of product functor. (Contributed by Zhi Wang, 29-Sep-2025.)

Hypotheses

Ref	Expression
fuco2eld.w	⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅))
fuco2eld.u	⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
fuco2eld.k	⊢ (𝜑 → 𝐾𝑆𝐿)
fuco2eld.f	⊢ (𝜑 → 𝐹𝑅𝐺)

Assertion

Ref	Expression
fuco2eld	⊢ (𝜑 → 𝑈 ∈ 𝑊)

Proof of Theorem fuco2eld

Step	Hyp	Ref	Expression
1		fuco2eld.k	. . 3 ⊢ (𝜑 → 𝐾𝑆𝐿)
2		fuco2eld.f	. . 3 ⊢ (𝜑 → 𝐹𝑅𝐺)
3		fuco2el 49897	. . 3 ⊢ (⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩ ∈ (𝑆 × 𝑅) ↔ (𝐾𝑆𝐿 ∧ 𝐹𝑅𝐺))
4	1, 2, 3	sylanbrc 592	. 2 ⊢ (𝜑 → ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩ ∈ (𝑆 × 𝑅))
5		fuco2eld.u	. 2 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
6		fuco2eld.w	. 2 ⊢ (𝜑 → 𝑊 = (𝑆 × 𝑅))
7	4, 5, 6	3eltr4d 2876	1 ⊢ (𝜑 → 𝑈 ∈ 𝑊)

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ⟨cop 4587   class class class wbr 5099   × cxp 5643

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-ext 2733  ax-sep 5245  ax-pr 5389

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-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  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-br 5100  df-opab 5162  df-xp 5651

This theorem is referenced by:  fuco11  49911  fuco11cl  49912  fuco21  49921