Theorem eliinid 45148

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 4941	. . . 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 3221	. 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 2111  ∀wral 3047  ∩ ciin 4937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-iin 4939

This theorem is referenced by:  iinssiin  45166  fnlimfvre  45712  smflimlem2  46810  smflimmpt  46848  smfsuplem1  46849  smfsupmpt  46853  smfsupxr  46854  smfinflem  46855  smfinfmpt  46857  smflimsuplem4  46861  smflimsupmpt  46867  smfliminfmpt  46870  fsupdm  46880  finfdm  46884