Theorem iinss2 5061

Description: An indexed intersection is included in any of its members. (Contributed by FL, 15-Oct-2012.)

Assertion

Ref	Expression
iinss2	⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵)

Proof of Theorem iinss2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliin 5003	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
2	1	elv 3478	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3		rsp 3242	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵))
4	3	com12 32	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐵))
5	2, 4	biimtrid 241	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ 𝐵))
6	5	ssrdv 3989	1 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2104  ∀wral 3059  Vcvv 3472   ⊆ wss 3949  ∩ ciin 4999

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-v 3474  df-in 3956  df-ss 3966  df-iin 5001

This theorem is referenced by:  dmiin  5953  gruiin  10809  txtube  23366  iooiinicc  44555  iooiinioc  44569  meaiininclem  45502  smfsuplem1  45827  smfsuplem3  45829  smflimsuplem2  45837