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Theorem islocfin 21443
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 21433 . . . . 5 LocFin = (𝑗 ∈ Top ↦ {𝑦 ∣ ( 𝑗 = 𝑦 ∧ ∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
21dmmptss 5744 . . . 4 dom LocFin ⊆ Top
3 elfvdm 6333 . . . 4 (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ dom LocFin)
42, 3sseldi 3707 . . 3 (𝐴 ∈ (LocFin‘𝐽) → 𝐽 ∈ Top)
5 eqimss2 3764 . . . . . . . . . . 11 (𝑋 = 𝑦 𝑦𝑋)
6 sspwuni 4719 . . . . . . . . . . 11 (𝑦 ⊆ 𝒫 𝑋 𝑦𝑋)
75, 6sylibr 224 . . . . . . . . . 10 (𝑋 = 𝑦𝑦 ⊆ 𝒫 𝑋)
8 selpw 4273 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝒫 𝑋𝑦 ⊆ 𝒫 𝑋)
97, 8sylibr 224 . . . . . . . . 9 (𝑋 = 𝑦𝑦 ∈ 𝒫 𝒫 𝑋)
109adantr 472 . . . . . . . 8 ((𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) → 𝑦 ∈ 𝒫 𝒫 𝑋)
1110abssi 3783 . . . . . . 7 {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ⊆ 𝒫 𝒫 𝑋
12 islocfin.1 . . . . . . . . 9 𝑋 = 𝐽
1312topopn 20834 . . . . . . . 8 (𝐽 ∈ Top → 𝑋𝐽)
14 pwexg 4955 . . . . . . . 8 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
15 pwexg 4955 . . . . . . . 8 (𝒫 𝑋 ∈ V → 𝒫 𝒫 𝑋 ∈ V)
1613, 14, 153syl 18 . . . . . . 7 (𝐽 ∈ Top → 𝒫 𝒫 𝑋 ∈ V)
17 ssexg 4912 . . . . . . 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 698 . . . . . 6 (𝐽 ∈ Top → {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ∈ V)
19 unieq 4552 . . . . . . . . . . 11 (𝑗 = 𝐽 𝑗 = 𝐽)
2019, 12syl6eqr 2776 . . . . . . . . . 10 (𝑗 = 𝐽 𝑗 = 𝑋)
2120eqeq1d 2726 . . . . . . . . 9 (𝑗 = 𝐽 → ( 𝑗 = 𝑦𝑋 = 𝑦))
22 rexeq 3242 . . . . . . . . . 10 (𝑗 = 𝐽 → (∃𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
2320, 22raleqbidv 3255 . . . . . . . . 9 (𝑗 = 𝐽 → (∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
2421, 23anbi12d 749 . . . . . . . 8 (𝑗 = 𝐽 → (( 𝑗 = 𝑦 ∧ ∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) ↔ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
2524abbidv 2843 . . . . . . 7 (𝑗 = 𝐽 → {𝑦 ∣ ( 𝑗 = 𝑦 ∧ ∀𝑥 𝑗𝑛𝑗 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} = {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
2625, 1fvmptg 6394 . . . . . 6 ((𝐽 ∈ Top ∧ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ∈ V) → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
2718, 26mpdan 705 . . . . 5 (𝐽 ∈ Top → (LocFin‘𝐽) = {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))})
2827eleq2d 2789 . . . 4 (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ 𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))}))
29 elex 3316 . . . . . 6 (𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} → 𝐴 ∈ V)
3029adantl 473 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))}) → 𝐴 ∈ V)
31 simpr 479 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
32 islocfin.2 . . . . . . . . . 10 𝑌 = 𝐴
3331, 32syl6eq 2774 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋 = 𝐴)
3413adantr 472 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝑋𝐽)
3533, 34eqeltrrd 2804 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴𝐽)
36 elex 3316 . . . . . . . 8 ( 𝐴𝐽 𝐴 ∈ V)
3735, 36syl 17 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
38 uniexb 7090 . . . . . . 7 (𝐴 ∈ V ↔ 𝐴 ∈ V)
3937, 38sylibr 224 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌) → 𝐴 ∈ V)
4039adantrr 755 . . . . 5 ((𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))) → 𝐴 ∈ V)
41 unieq 4552 . . . . . . . . 9 (𝑦 = 𝐴 𝑦 = 𝐴)
4241, 32syl6eqr 2776 . . . . . . . 8 (𝑦 = 𝐴 𝑦 = 𝑌)
4342eqeq2d 2734 . . . . . . 7 (𝑦 = 𝐴 → (𝑋 = 𝑦𝑋 = 𝑌))
44 rabeq 3296 . . . . . . . . . . 11 (𝑦 = 𝐴 → {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} = {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅})
4544eleq1d 2788 . . . . . . . . . 10 (𝑦 = 𝐴 → ({𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin ↔ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
4645anbi2d 742 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
4746rexbidv 3154 . . . . . . . 8 (𝑦 = 𝐴 → (∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
4847ralbidv 3088 . . . . . . 7 (𝑦 = 𝐴 → (∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
4943, 48anbi12d 749 . . . . . 6 (𝑦 = 𝐴 → ((𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
5049elabg 3456 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
5130, 40, 50pm5.21nd 979 . . . 4 (𝐽 ∈ Top → (𝐴 ∈ {𝑦 ∣ (𝑋 = 𝑦 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝑦 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))} ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
5228, 51bitrd 268 . . 3 (𝐽 ∈ Top → (𝐴 ∈ (LocFin‘𝐽) ↔ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
534, 52biadan2 677 . 2 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
54 3anass 1081 . 2 ((𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)) ↔ (𝐽 ∈ Top ∧ (𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))))
5553, 54bitr4i 267 1 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  {cab 2710  wne 2896  wral 3014  wrex 3015  {crab 3018  Vcvv 3304  cin 3679  wss 3680  c0 4023  𝒫 cpw 4266   cuni 4544  dom cdm 5218  cfv 6001  Fincfn 8072  Topctop 20821  LocFinclocfin 21430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fv 6009  df-top 20822  df-locfin 21433
This theorem is referenced by:  finlocfin  21446  locfintop  21447  locfinbas  21448  locfinnei  21449  lfinun  21451  dissnlocfin  21455  locfindis  21456  locfincf  21457  locfinreflem  30137  locfinref  30138
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