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Theorem islocfin 22101
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 22091 . . . 4 LocFin = (𝑗 ∈ Top ↦ {𝑦 ∣ ( 𝑗 = 𝑦 ∧ ∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
21mptrcl 6753 . . 3 (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
3 eqimss2 4003 . . . . . . . . . . 11 (𝑋 = 𝑦 𝑦𝑋)
4 sspwuni 4998 . . . . . . . . . . 11 (𝑦 ⊆ 𝒫 𝑋 𝑦𝑋)
53, 4sylibr 236 . . . . . . . . . 10 (𝑋 = 𝑦𝑦 ⊆ 𝒫 𝑋)
6 velpw 4520 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝒫 𝑋𝑦 ⊆ 𝒫 𝑋)
75, 6sylibr 236 . . . . . . . . 9 (𝑋 = 𝑦𝑦 ∈ 𝒫 𝒫 𝑋)
87adantr 483 . . . . . . . 8 ((𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) → 𝑦 ∈ 𝒫 𝒫 𝑋)
98abssi 4025 . . . . . . 7 {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ⊆ 𝒫 𝒫 𝑋
10 islocfin.1 . . . . . . . . 9 𝑋 = 𝐽
1110topopn 21490 . . . . . . . 8 (𝐽 ∈ Top → 𝑋𝐽)
12 pwexg 5255 . . . . . . . 8 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
13 pwexg 5255 . . . . . . . 8 (𝒫 𝑋 ∈ V → 𝒫 𝒫 𝑋 ∈ V)
1411, 12, 133syl 18 . . . . . . 7 (𝐽 ∈ Top → 𝒫 𝒫 𝑋 ∈ V)
15 ssexg 5203 . . . . . . 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 589 . . . . . 6 (𝐽 ∈ Top → {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ∈ V)
17 unieq 4825 . . . . . . . . . . 11 (𝑗 = 𝐽 𝑗 = 𝐽)
1817, 10syl6eqr 2873 . . . . . . . . . 10 (𝑗 = 𝐽 𝑗 = 𝑋)
1918eqeq1d 2822 . . . . . . . . 9 (𝑗 = 𝐽 → ( 𝑗 = 𝑦𝑋 = 𝑦))
20 rexeq 3393 . . . . . . . . . 10 (𝑗 = 𝐽 → (∃𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
2118, 20raleqbidv 3388 . . . . . . . . 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 2884 . . . . . . 7 (𝑗 = 𝐽 → {𝑦 ∣ ( 𝑗 = 𝑦 ∧ ∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} = {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
2423, 1fvmptg 6742 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ∈ V) → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
2516, 24mpdan 685 . . . . 5 (𝐽 ∈ Top → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
2625eleq2d 2896 . . . 4 (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ 𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))}))
27 elex 3491 . . . . . 6 (𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} → 𝐴 ∈ V)
2827adantl 484 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))}) → 𝐴 ∈ V)
29 simpr 487 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
30 islocfin.2 . . . . . . . . . 10 𝑌 = 𝐴
3129, 30syl6eq 2871 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = 𝐴)
3211adantr 483 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋𝐽)
3331, 32eqeltrrd 2912 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴𝐽)
3433elexd 3493 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
35 uniexb 7464 . . . . . . 7 (𝐴 ∈ V ↔ 𝐴 ∈ V)
3634, 35sylibr 236 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
3736adantrr 715 . . . . 5 ((𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))) → 𝐴 ∈ V)
38 unieq 4825 . . . . . . . . 9 (𝑦 = 𝐴 𝑦 = 𝐴)
3938, 30syl6eqr 2873 . . . . . . . 8 (𝑦 = 𝐴 𝑦 = 𝑌)
4039eqeq2d 2831 . . . . . . 7 (𝑦 = 𝐴 → (𝑋 = 𝑦𝑋 = 𝑌))
41 rabeq 3462 . . . . . . . . . . 11 (𝑦 = 𝐴 → {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} = {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅})
4241eleq1d 2895 . . . . . . . . . 10 (𝑦 = 𝐴 → ({𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin ↔ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
4342anbi2d 630 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
4443rexbidv 3284 . . . . . . . 8 (𝑦 = 𝐴 → (∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
4544ralbidv 3184 . . . . . . 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 3646 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
4828, 37, 47pm5.21nd 800 . . . 4 (𝐽 ∈ Top → (𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
4926, 48bitrd 281 . . 3 (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
502, 49biadanii 820 . 2 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
51 3anass 1091 . 2 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
5250, 51bitr4i 280 1 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  {cab 2798  wne 3006  wral 3125  wrex 3126  {crab 3129  Vcvv 3473  cin 3912  wss 3913  c0 4269  𝒫 cpw 4515   cuni 4814  cfv 6331  Fincfn 8487  Topctop 21477  LocFinclocfin 22088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fv 6339  df-top 21478  df-locfin 22091
This theorem is referenced by:  finlocfin  22104  locfintop  22105  locfinbas  22106  locfinnei  22107  lfinun  22109  dissnlocfin  22113  locfindis  22114  locfincf  22115  locfinreflem  31115  locfinref  31116
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