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Theorem islocfin 21534
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 21524 . . . . 5 LocFin = (𝑗 ∈ Top ↦ {𝑦 ∣ ( 𝑗 = 𝑦 ∧ ∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
21dmmptss 5845 . . . 4 dom LocFin ⊆ Top
3 elfvdm 6440 . . . 4 (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ dom LocFin)
42, 3sseldi 3796 . . 3 (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
5 eqimss2 3855 . . . . . . . . . . 11 (𝑋 = 𝑦 𝑦𝑋)
6 sspwuni 4803 . . . . . . . . . . 11 (𝑦 ⊆ 𝒫 𝑋 𝑦𝑋)
75, 6sylibr 225 . . . . . . . . . 10 (𝑋 = 𝑦𝑦 ⊆ 𝒫 𝑋)
8 selpw 4358 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝒫 𝑋𝑦 ⊆ 𝒫 𝑋)
97, 8sylibr 225 . . . . . . . . 9 (𝑋 = 𝑦𝑦 ∈ 𝒫 𝒫 𝑋)
109adantr 468 . . . . . . . 8 ((𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) → 𝑦 ∈ 𝒫 𝒫 𝑋)
1110abssi 3874 . . . . . . 7 {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ⊆ 𝒫 𝒫 𝑋
12 islocfin.1 . . . . . . . . 9 𝑋 = 𝐽
1312topopn 20924 . . . . . . . 8 (𝐽 ∈ Top → 𝑋𝐽)
14 pwexg 5048 . . . . . . . 8 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
15 pwexg 5048 . . . . . . . 8 (𝒫 𝑋 ∈ V → 𝒫 𝒫 𝑋 ∈ V)
1613, 14, 153syl 18 . . . . . . 7 (𝐽 ∈ Top → 𝒫 𝒫 𝑋 ∈ V)
17 ssexg 4999 . . . . . . 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)
1811, 16, 17sylancr 577 . . . . . 6 (𝐽 ∈ Top → {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ∈ V)
19 unieq 4638 . . . . . . . . . . 11 (𝑗 = 𝐽 𝑗 = 𝐽)
2019, 12syl6eqr 2858 . . . . . . . . . 10 (𝑗 = 𝐽 𝑗 = 𝑋)
2120eqeq1d 2808 . . . . . . . . 9 (𝑗 = 𝐽 → ( 𝑗 = 𝑦𝑋 = 𝑦))
22 rexeq 3328 . . . . . . . . . 10 (𝑗 = 𝐽 → (∃𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
2320, 22raleqbidv 3341 . . . . . . . . 9 (𝑗 = 𝐽 → (∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
2421, 23anbi12d 618 . . . . . . . 8 (𝑗 = 𝐽 → (( 𝑗 = 𝑦 ∧ ∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) ↔ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
2524abbidv 2925 . . . . . . 7 (𝑗 = 𝐽 → {𝑦 ∣ ( 𝑗 = 𝑦 ∧ ∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} = {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
2625, 1fvmptg 6501 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ∈ V) → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
2718, 26mpdan 670 . . . . 5 (𝐽 ∈ Top → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
2827eleq2d 2871 . . . 4 (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ 𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))}))
29 elex 3406 . . . . . 6 (𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} → 𝐴 ∈ V)
3029adantl 469 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))}) → 𝐴 ∈ V)
31 simpr 473 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
32 islocfin.2 . . . . . . . . . 10 𝑌 = 𝐴
3331, 32syl6eq 2856 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = 𝐴)
3413adantr 468 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋𝐽)
3533, 34eqeltrrd 2886 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴𝐽)
36 elex 3406 . . . . . . . 8 ( 𝐴𝐽 𝐴 ∈ V)
3735, 36syl 17 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
38 uniexb 7203 . . . . . . 7 (𝐴 ∈ V ↔ 𝐴 ∈ V)
3937, 38sylibr 225 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
4039adantrr 699 . . . . 5 ((𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))) → 𝐴 ∈ V)
41 unieq 4638 . . . . . . . . 9 (𝑦 = 𝐴 𝑦 = 𝐴)
4241, 32syl6eqr 2858 . . . . . . . 8 (𝑦 = 𝐴 𝑦 = 𝑌)
4342eqeq2d 2816 . . . . . . 7 (𝑦 = 𝐴 → (𝑋 = 𝑦𝑋 = 𝑌))
44 rabeq 3382 . . . . . . . . . . 11 (𝑦 = 𝐴 → {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} = {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅})
4544eleq1d 2870 . . . . . . . . . 10 (𝑦 = 𝐴 → ({𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin ↔ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
4645anbi2d 616 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
4746rexbidv 3240 . . . . . . . 8 (𝑦 = 𝐴 → (∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
4847ralbidv 3174 . . . . . . 7 (𝑦 = 𝐴 → (∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
4943, 48anbi12d 618 . . . . . 6 (𝑦 = 𝐴 → ((𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
5049elabg 3546 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
5130, 40, 50pm5.21nd 827 . . . 4 (𝐽 ∈ Top → (𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
5228, 51bitrd 270 . . 3 (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
534, 52biadan2 844 . 2 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
54 3anass 1109 . 2 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
5553, 54bitr4i 269 1 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  {cab 2792  wne 2978  wral 3096  wrex 3097  {crab 3100  Vcvv 3391  cin 3768  wss 3769  c0 4116  𝒫 cpw 4351   cuni 4630  dom cdm 5311  cfv 6101  Fincfn 8192  Topctop 20911  LocFinclocfin 21521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fv 6109  df-top 20912  df-locfin 21524
This theorem is referenced by:  finlocfin  21537  locfintop  21538  locfinbas  21539  locfinnei  21540  lfinun  21542  dissnlocfin  21546  locfindis  21547  locfincf  21548  locfinreflem  30232  locfinref  30233
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