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Theorem fneint 32468
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 {𝑥𝐵𝑃𝑥} ⊆ {𝑥𝐴𝑃𝑥})
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.
StepHypRef Expression
1 eleq2 2719 . . . . 5 (𝑥 = 𝑦 → (𝑃𝑥𝑃𝑦))
21elrab 3396 . . . 4 (𝑦 ∈ {𝑥𝐴𝑃𝑥} ↔ (𝑦𝐴𝑃𝑦))
3 fnessex 32466 . . . . . . 7 ((𝐴Fne𝐵𝑦𝐴𝑃𝑦) → ∃𝑧𝐵 (𝑃𝑧𝑧𝑦))
433expb 1285 . . . . . 6 ((𝐴Fne𝐵 ∧ (𝑦𝐴𝑃𝑦)) → ∃𝑧𝐵 (𝑃𝑧𝑧𝑦))
5 eleq2 2719 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑃𝑥𝑃𝑧))
65intminss 4535 . . . . . . . . 9 ((𝑧𝐵𝑃𝑧) → {𝑥𝐵𝑃𝑥} ⊆ 𝑧)
7 sstr 3644 . . . . . . . . 9 (( {𝑥𝐵𝑃𝑥} ⊆ 𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
86, 7sylan 487 . . . . . . . 8 (((𝑧𝐵𝑃𝑧) ∧ 𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
98expl 647 . . . . . . 7 (𝑧𝐵 → ((𝑃𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦))
109rexlimiv 3056 . . . . . 6 (∃𝑧𝐵 (𝑃𝑧𝑧𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
114, 10syl 17 . . . . 5 ((𝐴Fne𝐵 ∧ (𝑦𝐴𝑃𝑦)) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
1211ex 449 . . . 4 (𝐴Fne𝐵 → ((𝑦𝐴𝑃𝑦) → {𝑥𝐵𝑃𝑥} ⊆ 𝑦))
132, 12syl5bi 232 . . 3 (𝐴Fne𝐵 → (𝑦 ∈ {𝑥𝐴𝑃𝑥} → {𝑥𝐵𝑃𝑥} ⊆ 𝑦))
1413ralrimiv 2994 . 2 (𝐴Fne𝐵 → ∀𝑦 ∈ {𝑥𝐴𝑃𝑥} {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
15 ssint 4525 . 2 ( {𝑥𝐵𝑃𝑥} ⊆ {𝑥𝐴𝑃𝑥} ↔ ∀𝑦 ∈ {𝑥𝐴𝑃𝑥} {𝑥𝐵𝑃𝑥} ⊆ 𝑦)
1614, 15sylibr 224 1 (𝐴Fne𝐵 {𝑥𝐵𝑃𝑥} ⊆ {𝑥𝐴𝑃𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2030  wral 2941  wrex 2942  {crab 2945  wss 3607   cint 4507   class class class wbr 4685  Fnecfne 32456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topgen 16151  df-fne 32457
This theorem is referenced by: (None)
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