Theorem iinss 5014

Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iinss	⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

Distinct variable group:   𝑥,𝐶

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

Proof of Theorem iinss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliin 4953	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
2	1	elv 3447	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3		ssel 3929	. . . . 5 ⊢ (𝐵 ⊆ 𝐶 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
4	3	reximi 3076	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
5		r19.36v 3166	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
6	4, 5	syl 17	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶))
7	2, 6	biimtrid 242	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ 𝐶))
8	7	ssrdv 3941	1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ⊆ wss 3903  ∩ ciin 4949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-v 3444  df-ss 3920  df-iin 4951

This theorem is referenced by:  riinn0  5040  reliin  5774  cnviin  6252  iiner  8738  scott0  9810  cfslb  10188  ptbasfi  23537  iscmet3  25261  fnemeet1  36579  pmapglb2N  40144  pmapglb2xN  40145  iinssd  45487  iooiinicc  45899  iooiinioc  45913  meaiininclem  46841  iinhoiicclem  47028  smflim  47132  smflimsuplem7  47181  iinglb  49178  iineqconst2  49180  iinfssc  49413