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Theorem pwfir 9176
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 5690 . . 3 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵)
2 relopab 5825 . . . . 5 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
3 dmopabss 5919 . . . . 5 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ⊆ 𝒫 𝐵
4 relssres 6023 . . . . 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 691 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
65rneqi 5937 . . 3 ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
7 rnopab 5954 . . . 4 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
8 eleq1 2822 . . . . . . . . 9 ({𝑦} = 𝑥 → ({𝑦} ∈ 𝒫 𝐵𝑥 ∈ 𝒫 𝐵))
98biimparc 481 . . . . . . . 8 ((𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → {𝑦} ∈ 𝒫 𝐵)
10 vex 3479 . . . . . . . . 9 𝑦 ∈ V
1110snelpw 5446 . . . . . . . 8 (𝑦𝐵 ↔ {𝑦} ∈ 𝒫 𝐵)
129, 11sylibr 233 . . . . . . 7 ((𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → 𝑦𝐵)
1312exlimiv 1934 . . . . . 6 (∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → 𝑦𝐵)
14 snelpwi 5444 . . . . . . . 8 (𝑦𝐵 → {𝑦} ∈ 𝒫 𝐵)
15 eqid 2733 . . . . . . . 8 {𝑦} = {𝑦}
16 eqeq2 2745 . . . . . . . . 9 (𝑥 = {𝑦} → ({𝑦} = 𝑥 ↔ {𝑦} = {𝑦}))
1716rspcev 3613 . . . . . . . 8 (({𝑦} ∈ 𝒫 𝐵 ∧ {𝑦} = {𝑦}) → ∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥)
1814, 15, 17sylancl 587 . . . . . . 7 (𝑦𝐵 → ∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥)
19 df-rex 3072 . . . . . . 7 (∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥))
2018, 19sylib 217 . . . . . 6 (𝑦𝐵 → ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥))
2113, 20impbii 208 . . . . 5 (∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) ↔ 𝑦𝐵)
2221abbii 2803 . . . 4 {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = {𝑦𝑦𝐵}
23 abid2 2872 . . . 4 {𝑦𝑦𝐵} = 𝐵
247, 22, 233eqtri 2765 . . 3 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = 𝐵
251, 6, 243eqtri 2765 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) = 𝐵
26 funopab 6584 . . . 4 (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥))
27 mosneq 4844 . . . . 5 ∃*𝑦{𝑦} = 𝑥
2827moani 2548 . . . 4 ∃*𝑦(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)
2926, 28mpgbir 1802 . . 3 Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
30 imafi 9175 . . 3 ((Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ∧ 𝒫 𝐵 ∈ Fin) → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) ∈ Fin)
3129, 30mpan 689 . 2 (𝒫 𝐵 ∈ Fin → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) ∈ Fin)
3225, 31eqeltrrid 2839 1 (𝒫 𝐵 ∈ Fin → 𝐵 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  ∃*wmo 2533  {cab 2710  wrex 3071  wss 3949  𝒫 cpw 4603  {csn 4629  {copab 5211  dom cdm 5677  ran crn 5678  cres 5679  cima 5680  Rel wrel 5682  Fun wfun 6538  Fincfn 8939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-en 8940  df-fin 8943
This theorem is referenced by:  pwfi  9178
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