Theorem eliinid 45116

Description: Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Assertion

Ref	Expression
eliinid	⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem eliinid

Step	Hyp	Ref	Expression
1		simpl 482	. . 3 ⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶)
2		eliin 4996	. . . 4 ⊢ (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
3	2	adantr 480	. . 3 ⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶))
4	1, 3	mpbid 232	. 2 ⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶)
5		rspa 3248	. 2 ⊢ ((∀𝑥 ∈ 𝐵 𝐴 ∈ 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶)
6	4, 5	sylancom 588	1 ⊢ ((𝐴 ∈ ∩ 𝑥 ∈ 𝐵 𝐶 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∀wral 3061  ∩ ciin 4992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-iin 4994

This theorem is referenced by:  iinssiin  45134  fnlimfvre  45689  smflimlem2  46787  smflimmpt  46825  smfsuplem1  46826  smfsupmpt  46830  smfsupxr  46831  smfinflem  46832  smfinfmpt  46834  smflimsuplem4  46838  smflimsupmpt  46844  smfliminfmpt  46847  fsupdm  46857  finfdm  46861