Theorem fuco2el 48967

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

Assertion

Ref	Expression
fuco2el	⊢ (⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩ ∈ (𝑆 × 𝑅) ↔ (𝐾𝑆𝐿 ∧ 𝐹𝑅𝐺))

Proof of Theorem fuco2el

Step	Hyp	Ref	Expression
1		opelxp 5703	. 2 ⊢ (⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩ ∈ (𝑆 × 𝑅) ↔ (⟨𝐾, 𝐿⟩ ∈ 𝑆 ∧ ⟨𝐹, 𝐺⟩ ∈ 𝑅))
2		df-br 5126	. . 3 ⊢ (𝐾𝑆𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ 𝑆)
3		df-br 5126	. . 3 ⊢ (𝐹𝑅𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ 𝑅)
4	2, 3	anbi12i 628	. 2 ⊢ ((𝐾𝑆𝐿 ∧ 𝐹𝑅𝐺) ↔ (⟨𝐾, 𝐿⟩ ∈ 𝑆 ∧ ⟨𝐹, 𝐺⟩ ∈ 𝑅))
5	1, 4	bitr4i 278	1 ⊢ (⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩ ∈ (𝑆 × 𝑅) ↔ (𝐾𝑆𝐿 ∧ 𝐹𝑅𝐺))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2107  ⟨cop 4614   class class class wbr 5125   × cxp 5665

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-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

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-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-br 5126  df-opab 5188  df-xp 5673

This theorem is referenced by:  fuco2eld  48968  fuco2eld3  48970