Theorem fneint 36744

Description: If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.)

Assertion

Ref	Expression
fneint	⊢ (𝐴Fne𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ ∩ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑃

Proof of Theorem fneint

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2w 2853	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
2	1	elrab 3659	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ↔ (𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦))
3		fnessex 36742	. . . . . . 7 ⊢ ((𝐴Fne𝐵 ∧ 𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦) → ∃𝑧 ∈ 𝐵 (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))
4	3	3expb 1136	. . . . . 6 ⊢ ((𝐴Fne𝐵 ∧ (𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦)) → ∃𝑧 ∈ 𝐵 (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))
5		eleq2w 2853	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑧))
6	5	intminss 4940	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑃 ∈ 𝑧) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑧)
7		sstr 3953	. . . . . . . . 9 ⊢ ((∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
8	6, 7	sylan 591	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝐵 ∧ 𝑃 ∈ 𝑧) ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
9	8	expl 462	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → ((𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦))
10	9	rexlimiv 3165	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐵 (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
11	4, 10	syl 18	. . . . 5 ⊢ ((𝐴Fne𝐵 ∧ (𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦)) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
12	11	ex 417	. . . 4 ⊢ (𝐴Fne𝐵 → ((𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦))
13	2, 12	biimtrid 245	. . 3 ⊢ (𝐴Fne𝐵 → (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦))
14	13	ralrimiv 3162	. 2 ⊢ (𝐴Fne𝐵 → ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
15		ssint 4930	. 2 ⊢ (∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ ∩ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
16	14, 15	sylibr 237	1 ⊢ (𝐴Fne𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ ∩ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥})

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3423   ⊆ wss 3913  ∩ cint 4913   class class class wbr 5110  Fnecfne 36732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4914  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6490  df-fun 6536  df-fv 6542  df-topgen 17492  df-fne 36733

This theorem is referenced by: (None)