Theorem chnrin2 48111

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 48110	. . . 4 ⊢ (𝑛 ∈ ((𝑅 ∩ < ) Chain 𝐵) → (𝑛 ∈ (𝑅 Chain 𝐵) ∧ 𝑛 ∈ ( < Chain 𝐵)))
2		chnrin 48109	. . . 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 4096	. . 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 2760	1 ⊢ ((𝑅 ∩ < ) Chain 𝐵) = ((𝑅 Chain 𝐵) ∩ ( < Chain 𝐵))

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ∩ cin 4092   Chain cchn 18805

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-dm 5665  df-iota 6509  df-fv 6563  df-chn 18806

This theorem is used by: (None)