Theorem fneint 35768

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 2812	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
2	1	elrab 3680	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ↔ (𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦))
3		fnessex 35766	. . . . . . 7 ⊢ ((𝐴Fne𝐵 ∧ 𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦) → ∃𝑧 ∈ 𝐵 (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))
4	3	3expb 1118	. . . . . 6 ⊢ ((𝐴Fne𝐵 ∧ (𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦)) → ∃𝑧 ∈ 𝐵 (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))
5		eleq2w 2812	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑧))
6	5	intminss 4972	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑃 ∈ 𝑧) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑧)
7		sstr 3986	. . . . . . . . 9 ⊢ ((∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
8	6, 7	sylan 579	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝐵 ∧ 𝑃 ∈ 𝑧) ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
9	8	expl 457	. . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → ((𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦))
10	9	rexlimiv 3143	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐵 (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
11	4, 10	syl 17	. . . . 5 ⊢ ((𝐴Fne𝐵 ∧ (𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦)) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
12	11	ex 412	. . . 4 ⊢ (𝐴Fne𝐵 → ((𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦))
13	2, 12	biimtrid 241	. . 3 ⊢ (𝐴Fne𝐵 → (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦))
14	13	ralrimiv 3140	. 2 ⊢ (𝐴Fne𝐵 → ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
15		ssint 4962	. 2 ⊢ (∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ ∩ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
16	14, 15	sylibr 233	1 ⊢ (𝐴Fne𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ ∩ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥})

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2099  ∀wral 3056  ∃wrex 3065  {crab 3427   ⊆ wss 3944  ∩ cint 4944   class class class wbr 5142  Fnecfne 35756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-topgen 17416  df-fne 35757

This theorem is referenced by: (None)