Theorem iinss2 5021

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 4960	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
2	1	elv 3452	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3		rsp 3225	. . . 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 242	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ 𝐵))
6	5	ssrdv 3952	1 ⊢ (𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ⊆ wss 3914  ∩ ciin 4956

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 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-v 3449  df-ss 3931  df-iin 4958

This theorem is referenced by:  dmiin  5917  gruiin  10763  txtube  23527  iooiinicc  45540  iooiinioc  45554  meaiininclem  46484  smfsuplem1  46809  smfsuplem3  46811  smflimsuplem2  46819  iinfssc  49046  iinfsubc  49047