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Theorem islocfin 23541
Description: The statement "is a locally finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypotheses
Ref Expression
islocfin.1 𝑋 = 𝐽
islocfin.2 𝑌 = 𝐴
Assertion
Ref Expression
islocfin (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
Distinct variable groups:   𝑛,𝑠,𝑥,𝐴   𝑛,𝐽,𝑥   𝑥,𝑋
Allowed substitution hints:   𝐽(𝑠)   𝑋(𝑛,𝑠)   𝑌(𝑥,𝑛,𝑠)

Proof of Theorem islocfin
Dummy variables 𝑗 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-locfin 23531 . . . 4 LocFin = (𝑗 ∈ Top ↦ {𝑦 ∣ ( 𝑗 = 𝑦 ∧ ∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
21mptrcl 7025 . . 3 (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
3 eqimss2 4055 . . . . . . . . . . 11 (𝑋 = 𝑦 𝑦𝑋)
4 sspwuni 5105 . . . . . . . . . . 11 (𝑦 ⊆ 𝒫 𝑋 𝑦𝑋)
53, 4sylibr 234 . . . . . . . . . 10 (𝑋 = 𝑦𝑦 ⊆ 𝒫 𝑋)
6 velpw 4610 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝒫 𝑋𝑦 ⊆ 𝒫 𝑋)
75, 6sylibr 234 . . . . . . . . 9 (𝑋 = 𝑦𝑦 ∈ 𝒫 𝒫 𝑋)
87adantr 480 . . . . . . . 8 ((𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) → 𝑦 ∈ 𝒫 𝒫 𝑋)
98abssi 4080 . . . . . . 7 {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ⊆ 𝒫 𝒫 𝑋
10 islocfin.1 . . . . . . . . 9 𝑋 = 𝐽
1110topopn 22928 . . . . . . . 8 (𝐽 ∈ Top → 𝑋𝐽)
12 pwexg 5384 . . . . . . . 8 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
13 pwexg 5384 . . . . . . . 8 (𝒫 𝑋 ∈ V → 𝒫 𝒫 𝑋 ∈ V)
1411, 12, 133syl 18 . . . . . . 7 (𝐽 ∈ Top → 𝒫 𝒫 𝑋 ∈ V)
15 ssexg 5329 . . . . . . 7 (({𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ⊆ 𝒫 𝒫 𝑋 ∧ 𝒫 𝒫 𝑋 ∈ V) → {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ∈ V)
169, 14, 15sylancr 587 . . . . . 6 (𝐽 ∈ Top → {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ∈ V)
17 unieq 4923 . . . . . . . . . . 11 (𝑗 = 𝐽 𝑗 = 𝐽)
1817, 10eqtr4di 2793 . . . . . . . . . 10 (𝑗 = 𝐽 𝑗 = 𝑋)
1918eqeq1d 2737 . . . . . . . . 9 (𝑗 = 𝐽 → ( 𝑗 = 𝑦𝑋 = 𝑦))
20 rexeq 3320 . . . . . . . . . 10 (𝑗 = 𝐽 → (∃𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
2118, 20raleqbidv 3344 . . . . . . . . 9 (𝑗 = 𝐽 → (∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
2219, 21anbi12d 632 . . . . . . . 8 (𝑗 = 𝐽 → (( 𝑗 = 𝑦 ∧ ∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) ↔ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
2322abbidv 2806 . . . . . . 7 (𝑗 = 𝐽 → {𝑦 ∣ ( 𝑗 = 𝑦 ∧ ∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} = {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
2423, 1fvmptg 7014 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ∈ V) → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
2516, 24mpdan 687 . . . . 5 (𝐽 ∈ Top → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
2625eleq2d 2825 . . . 4 (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ 𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))}))
27 elex 3499 . . . . . 6 (𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} → 𝐴 ∈ V)
2827adantl 481 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))}) → 𝐴 ∈ V)
29 simpr 484 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
30 islocfin.2 . . . . . . . . . 10 𝑌 = 𝐴
3129, 30eqtrdi 2791 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = 𝐴)
3211adantr 480 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋𝐽)
3331, 32eqeltrrd 2840 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴𝐽)
3433elexd 3502 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
35 uniexb 7783 . . . . . . 7 (𝐴 ∈ V ↔ 𝐴 ∈ V)
3634, 35sylibr 234 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
3736adantrr 717 . . . . 5 ((𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))) → 𝐴 ∈ V)
38 unieq 4923 . . . . . . . . 9 (𝑦 = 𝐴 𝑦 = 𝐴)
3938, 30eqtr4di 2793 . . . . . . . 8 (𝑦 = 𝐴 𝑦 = 𝑌)
4039eqeq2d 2746 . . . . . . 7 (𝑦 = 𝐴 → (𝑋 = 𝑦𝑋 = 𝑌))
41 rabeq 3448 . . . . . . . . . . 11 (𝑦 = 𝐴 → {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} = {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅})
4241eleq1d 2824 . . . . . . . . . 10 (𝑦 = 𝐴 → ({𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin ↔ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
4342anbi2d 630 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
4443rexbidv 3177 . . . . . . . 8 (𝑦 = 𝐴 → (∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
4544ralbidv 3176 . . . . . . 7 (𝑦 = 𝐴 → (∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
4640, 45anbi12d 632 . . . . . 6 (𝑦 = 𝐴 → ((𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
4746elabg 3677 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
4828, 37, 47pm5.21nd 802 . . . 4 (𝐽 ∈ Top → (𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
4926, 48bitrd 279 . . 3 (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
502, 49biadanii 822 . 2 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
51 3anass 1094 . 2 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
5250, 51bitr4i 278 1 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912  cfv 6563  Fincfn 8984  Topctop 22915  LocFinclocfin 23528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-top 22916  df-locfin 23531
This theorem is referenced by:  finlocfin  23544  locfintop  23545  locfinbas  23546  locfinnei  23547  lfinun  23549  dissnlocfin  23553  locfindis  23554  locfincf  23555  locfinreflem  33801  locfinref  33802
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