Theorem fuco2el 49665

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 5668	. 2 ⊢ (⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩ ∈ (𝑆 × 𝑅) ↔ (⟨𝐾, 𝐿⟩ ∈ 𝑆 ∧ ⟨𝐹, 𝐺⟩ ∈ 𝑅))
2		df-br 5101	. . 3 ⊢ (𝐾𝑆𝐿 ↔ ⟨𝐾, 𝐿⟩ ∈ 𝑆)
3		df-br 5101	. . 3 ⊢ (𝐹𝑅𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ 𝑅)
4	2, 3	anbi12i 629	. 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 2114  ⟨cop 4588   class class class wbr 5100   × cxp 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638

This theorem is referenced by:  fuco2eld  49666  fuco2eld3  49668