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Theorem pwfir 9355
Description: If the power set of a set is finite, then the set itself is finite. (Contributed by BTernaryTau, 7-Sep-2024.)
Assertion
Ref Expression
pwfir (𝒫 𝐵 ∈ Fin → 𝐵 ∈ Fin)

Proof of Theorem pwfir
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ima 5698 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵)
2 relopab 5834 . . . . 5 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
3 dmopabss 5929 . . . . 5 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ⊆ 𝒫 𝐵
4 relssres 6040 . . . . 5 ((Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ∧ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ⊆ 𝒫 𝐵) → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)})
52, 3, 4mp2an 692 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
65rneqi 5948 . . 3 ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
7 rnopab 5965 . . . 4 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
8 eleq1 2829 . . . . . . . . 9 ({𝑦} = 𝑥 → ({𝑦} ∈ 𝒫 𝐵𝑥 ∈ 𝒫 𝐵))
98biimparc 479 . . . . . . . 8 ((𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → {𝑦} ∈ 𝒫 𝐵)
10 vex 3484 . . . . . . . . 9 𝑦 ∈ V
1110snelpw 5450 . . . . . . . 8 (𝑦𝐵 ↔ {𝑦} ∈ 𝒫 𝐵)
129, 11sylibr 234 . . . . . . 7 ((𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → 𝑦𝐵)
1312exlimiv 1930 . . . . . 6 (∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → 𝑦𝐵)
14 snelpwi 5448 . . . . . . . 8 (𝑦𝐵 → {𝑦} ∈ 𝒫 𝐵)
15 eqid 2737 . . . . . . . 8 {𝑦} = {𝑦}
16 eqeq2 2749 . . . . . . . . 9 (𝑥 = {𝑦} → ({𝑦} = 𝑥 ↔ {𝑦} = {𝑦}))
1716rspcev 3622 . . . . . . . 8 (({𝑦} ∈ 𝒫 𝐵 ∧ {𝑦} = {𝑦}) → ∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥)
1814, 15, 17sylancl 586 . . . . . . 7 (𝑦𝐵 → ∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥)
19 df-rex 3071 . . . . . . 7 (∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥))
2018, 19sylib 218 . . . . . 6 (𝑦𝐵 → ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥))
2113, 20impbii 209 . . . . 5 (∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) ↔ 𝑦𝐵)
2221abbii 2809 . . . 4 {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = {𝑦𝑦𝐵}
23 abid2 2879 . . . 4 {𝑦𝑦𝐵} = 𝐵
247, 22, 233eqtri 2769 . . 3 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = 𝐵
251, 6, 243eqtri 2769 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) = 𝐵
26 funopab 6601 . . . 4 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥))
27 mosneq 4842 . . . . 5 ∃*𝑦{𝑦} = 𝑥
2827moani 2553 . . . 4 ∃*𝑦(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)
2926, 28mpgbir 1799 . . 3 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
30 imafi 9353 . . 3 ((Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ∧ 𝒫 𝐵 ∈ Fin) → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) ∈ Fin)
3129, 30mpan 690 . 2 (𝒫 𝐵 ∈ Fin → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) ∈ Fin)
3225, 31eqeltrrid 2846 1 (𝒫 𝐵 ∈ Fin → 𝐵 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  ∃*wmo 2538  {cab 2714  wrex 3070  wss 3951  𝒫 cpw 4600  {csn 4626  {copab 5205  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  Rel wrel 5690  Fun wfun 6555  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-en 8986  df-dom 8987  df-fin 8989
This theorem is referenced by:  pwfi  9357
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