Theorem fneint 36530

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 2820	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
2	1	elrab 3634	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ↔ (𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦))
3		fnessex 36528	. . . . . . 7 ⊢ ((𝐴Fne𝐵 ∧ 𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦) → ∃𝑧 ∈ 𝐵 (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))
4	3	3expb 1121	. . . . . 6 ⊢ ((𝐴Fne𝐵 ∧ (𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦)) → ∃𝑧 ∈ 𝐵 (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦))
5		eleq2w 2820	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑧))
6	5	intminss 4916	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑃 ∈ 𝑧) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑧)
7		sstr 3930	. . . . . . . . 9 ⊢ ((∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
8	6, 7	sylan 581	. . . . . . . 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 3131	. . . . . 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 3128	. 2 ⊢ (𝐴Fne𝐵 → ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)
15		ssint 4906	. 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 2114  ∀wral 3051  ∃wrex 3061  {crab 3389   ⊆ wss 3889  ∩ cint 4889   class class class wbr 5085  Fnecfne 36518

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-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-topgen 17406  df-fne 36519

This theorem is referenced by: (None)