Theorem fneint 36366

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 2818	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
2	1	elrab 3671	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ↔ (𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦))
3		fnessex 36364	. . . . . . 7 ⊢ ((𝐴Fne𝐵 ∧ 𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦) → ∃𝑧 ∈ 𝐵 (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))
4	3	3expb 1120	. . . . . 6 ⊢ ((𝐴Fne𝐵 ∧ (𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦)) → ∃𝑧 ∈ 𝐵 (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))
5		eleq2w 2818	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑧))
6	5	intminss 4950	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑃 ∈ 𝑧) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑧)
7		sstr 3967	. . . . . . . . 9 ⊢ ((∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
8	6, 7	sylan 580	. . . . . . . 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 3134	. . . . . 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 242	. . 3 ⊢ (𝐴Fne𝐵 → (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦))
14	13	ralrimiv 3131	. 2 ⊢ (𝐴Fne𝐵 → ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
15		ssint 4940	. 2 ⊢ (∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ ∩ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
16	14, 15	sylibr 234	1 ⊢ (𝐴Fne𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ ∩ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥})

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  {crab 3415   ⊆ wss 3926  ∩ cint 4922   class class class wbr 5119  Fnecfne 36354

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-topgen 17457  df-fne 36355

This theorem is referenced by: (None)