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Theorem finlocfin 21246
 Description: A finite cover of a topological space is a locally finite cover. (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypotheses
Ref Expression
finlocfin.1 𝑋 = 𝐽
finlocfin.2 𝑌 = 𝐴
Assertion
Ref Expression
finlocfin ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐴 ∈ (LocFin‘𝐽))

Proof of Theorem finlocfin
Dummy variables 𝑛 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1059 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐽 ∈ Top)
2 simp3 1061 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
3 simpl1 1062 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝐽 ∈ Top)
4 finlocfin.1 . . . . . 6 𝑋 = 𝐽
54topopn 20643 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
63, 5syl 17 . . . 4 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝑋𝐽)
7 simpr 477 . . . 4 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝑥𝑋)
8 simpl2 1063 . . . . 5 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → 𝐴 ∈ Fin)
9 ssrab2 3671 . . . . 5 {𝑠𝐴 ∣ (𝑠𝑋) ≠ ∅} ⊆ 𝐴
10 ssfi 8132 . . . . 5 ((𝐴 ∈ Fin ∧ {𝑠𝐴 ∣ (𝑠𝑋) ≠ ∅} ⊆ 𝐴) → {𝑠𝐴 ∣ (𝑠𝑋) ≠ ∅} ∈ Fin)
118, 9, 10sylancl 693 . . . 4 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → {𝑠𝐴 ∣ (𝑠𝑋) ≠ ∅} ∈ Fin)
12 eleq2 2687 . . . . . 6 (𝑛 = 𝑋 → (𝑥𝑛𝑥𝑋))
13 ineq2 3791 . . . . . . . . 9 (𝑛 = 𝑋 → (𝑠𝑛) = (𝑠𝑋))
1413neeq1d 2849 . . . . . . . 8 (𝑛 = 𝑋 → ((𝑠𝑛) ≠ ∅ ↔ (𝑠𝑋) ≠ ∅))
1514rabbidv 3180 . . . . . . 7 (𝑛 = 𝑋 → {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} = {𝑠𝐴 ∣ (𝑠𝑋) ≠ ∅})
1615eleq1d 2683 . . . . . 6 (𝑛 = 𝑋 → ({𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin ↔ {𝑠𝐴 ∣ (𝑠𝑋) ≠ ∅} ∈ Fin))
1712, 16anbi12d 746 . . . . 5 (𝑛 = 𝑋 → ((𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin) ↔ (𝑥𝑋 ∧ {𝑠𝐴 ∣ (𝑠𝑋) ≠ ∅} ∈ Fin)))
1817rspcev 3298 . . . 4 ((𝑋𝐽 ∧ (𝑥𝑋 ∧ {𝑠𝐴 ∣ (𝑠𝑋) ≠ ∅} ∈ Fin)) → ∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
196, 7, 11, 18syl12anc 1321 . . 3 (((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) ∧ 𝑥𝑋) → ∃𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
2019ralrimiva 2961 . 2 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin))
21 finlocfin.2 . . 3 𝑌 = 𝐴
224, 21islocfin 21243 . 2 (𝐴 ∈ (LocFin‘𝐽) ↔ (𝐽 ∈ Top ∧ 𝑋 = 𝑌 ∧ ∀𝑥𝑋𝑛𝐽 (𝑥𝑛 ∧ {𝑠𝐴 ∣ (𝑠𝑛) ≠ ∅} ∈ Fin)))
231, 2, 20, 22syl3anbrc 1244 1 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ 𝑋 = 𝑌) → 𝐴 ∈ (LocFin‘𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907  ∃wrex 2908  {crab 2911   ∩ cin 3558   ⊆ wss 3559  ∅c0 3896  ∪ cuni 4407  ‘cfv 5852  Fincfn 7907  Topctop 20630  LocFinclocfin 21230 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-er 7694  df-en 7908  df-fin 7911  df-top 20631  df-locfin 21233 This theorem is referenced by:  locfincmp  21252  cmppcmp  29731
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