Theorem chnrin2 47908

Description: Distribution of chain class constructor over relation intersection. (Contributed by Ender Ting, 24-Jul-2026.)

Assertion

Ref	Expression
chnrin2	⊢ ((𝑅 ∩ < ) Chain 𝐵) = ((𝑅 Chain 𝐵) ∩ ( < Chain 𝐵))

Proof of Theorem chnrin2

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		chnrrin 47907	. . . 4 ⊢ (𝑛 ∈ ((𝑅 ∩ < ) Chain 𝐵) → (𝑛 ∈ (𝑅 Chain 𝐵) ∧ 𝑛 ∈ ( < Chain 𝐵)))
2		chnrin 47906	. . . 4 ⊢ ((𝑛 ∈ (𝑅 Chain 𝐵) ∧ 𝑛 ∈ ( < Chain 𝐵)) → 𝑛 ∈ ((𝑅 ∩ < ) Chain 𝐵))
3	1, 2	impbii 212	. . 3 ⊢ (𝑛 ∈ ((𝑅 ∩ < ) Chain 𝐵) ↔ (𝑛 ∈ (𝑅 Chain 𝐵) ∧ 𝑛 ∈ ( < Chain 𝐵)))
4		elin 4093	. . 3 ⊢ (𝑛 ∈ ((𝑅 Chain 𝐵) ∩ ( < Chain 𝐵)) ↔ (𝑛 ∈ (𝑅 Chain 𝐵) ∧ 𝑛 ∈ ( < Chain 𝐵)))
5	3, 4	bitr4i 281	. 2 ⊢ (𝑛 ∈ ((𝑅 ∩ < ) Chain 𝐵) ↔ 𝑛 ∈ ((𝑅 Chain 𝐵) ∩ ( < Chain 𝐵)))
6	5	eqriv 2757	1 ⊢ ((𝑅 ∩ < ) Chain 𝐵) = ((𝑅 Chain 𝐵) ∩ ( < Chain 𝐵))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ∩ cin 4089   Chain cchn 18779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-dm 5660  df-iota 6494  df-fv 6546  df-chn 18780

This theorem is used by: (None)