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Theorem unifpw 9393
Description: A set is the union of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
Assertion
Ref Expression
unifpw (𝒫 𝐴 ∩ Fin) = 𝐴

Proof of Theorem unifpw
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 inss1 4245 . . . . . 6 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
21unissi 4921 . . . . 5 (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
3 unipw 5461 . . . . 5 𝒫 𝐴 = 𝐴
42, 3sseqtri 4032 . . . 4 (𝒫 𝐴 ∩ Fin) ⊆ 𝐴
54sseli 3991 . . 3 (𝑎 (𝒫 𝐴 ∩ Fin) → 𝑎𝐴)
6 snelpwi 5454 . . . . . 6 (𝑎𝐴 → {𝑎} ∈ 𝒫 𝐴)
7 snfi 9082 . . . . . . 7 {𝑎} ∈ Fin
87a1i 11 . . . . . 6 (𝑎𝐴 → {𝑎} ∈ Fin)
96, 8elind 4210 . . . . 5 (𝑎𝐴 → {𝑎} ∈ (𝒫 𝐴 ∩ Fin))
10 elssuni 4942 . . . . 5 ({𝑎} ∈ (𝒫 𝐴 ∩ Fin) → {𝑎} ⊆ (𝒫 𝐴 ∩ Fin))
119, 10syl 17 . . . 4 (𝑎𝐴 → {𝑎} ⊆ (𝒫 𝐴 ∩ Fin))
12 snidg 4665 . . . 4 (𝑎𝐴𝑎 ∈ {𝑎})
1311, 12sseldd 3996 . . 3 (𝑎𝐴𝑎 (𝒫 𝐴 ∩ Fin))
145, 13impbii 209 . 2 (𝑎 (𝒫 𝐴 ∩ Fin) ↔ 𝑎𝐴)
1514eqriv 2732 1 (𝒫 𝐴 ∩ Fin) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  cin 3962  wss 3963  𝒫 cpw 4605  {csn 4631   cuni 4912  Fincfn 8984
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-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  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-pss 3983  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-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-om 7888  df-1o 8505  df-en 8985  df-fin 8988
This theorem is referenced by:  isacs5lem  18603  acsmapd  18612  acsmap2d  18613
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