Theorem pwfir 9173

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.

Step	Hyp	Ref	Expression
1		df-ima 5689	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵)
2		relopab 5823	. . . . 5 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
3		dmopabss 5917	. . . . 5 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ⊆ 𝒫 𝐵
4		relssres 6021	. . . . 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)})
5	2, 3, 4	mp2an 691	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
6	5	rneqi 5935	. . 3 ⊢ ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
7		rnopab 5952	. . . 4 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
8		eleq1 2822	. . . . . . . . 9 ⊢ ({𝑦} = 𝑥 → ({𝑦} ∈ 𝒫 𝐵 ↔ 𝑥 ∈ 𝒫 𝐵))
9	8	biimparc 481	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → {𝑦} ∈ 𝒫 𝐵)
10		vex 3479	. . . . . . . . 9 ⊢ 𝑦 ∈ V
11	10	snelpw 5445	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 ↔ {𝑦} ∈ 𝒫 𝐵)
12	9, 11	sylibr 233	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → 𝑦 ∈ 𝐵)
13	12	exlimiv 1934	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → 𝑦 ∈ 𝐵)
14		snelpwi 5443	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ∈ 𝒫 𝐵)
15		eqid 2733	. . . . . . . 8 ⊢ {𝑦} = {𝑦}
16		eqeq2 2745	. . . . . . . . 9 ⊢ (𝑥 = {𝑦} → ({𝑦} = 𝑥 ↔ {𝑦} = {𝑦}))
17	16	rspcev 3613	. . . . . . . 8 ⊢ (({𝑦} ∈ 𝒫 𝐵 ∧ {𝑦} = {𝑦}) → ∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥)
18	14, 15, 17	sylancl 587	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥)
19		df-rex 3072	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥))
20	18, 19	sylib 217	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥))
21	13, 20	impbii 208	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) ↔ 𝑦 ∈ 𝐵)
22	21	abbii 2803	. . . 4 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = {𝑦 ∣ 𝑦 ∈ 𝐵}
23		abid2 2872	. . . 4 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐵} = 𝐵
24	7, 22, 23	3eqtri 2765	. . 3 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = 𝐵
25	1, 6, 24	3eqtri 2765	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) = 𝐵
26		funopab 6581	. . . 4 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥))
27		mosneq 4843	. . . . 5 ⊢ ∃*𝑦{𝑦} = 𝑥
28	27	moani 2548	. . . 4 ⊢ ∃*𝑦(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)
29	26, 28	mpgbir 1802	. . 3 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
30		imafi 9172	. . 3 ⊢ ((Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ∧ 𝒫 𝐵 ∈ Fin) → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) ∈ Fin)
31	29, 30	mpan 689	. 2 ⊢ (𝒫 𝐵 ∈ Fin → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) ∈ Fin)
32	25, 31	eqeltrrid 2839	1 ⊢ (𝒫 𝐵 ∈ Fin → 𝐵 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  ∃*wmo 2533  {cab 2710  ∃wrex 3071   ⊆ wss 3948  𝒫 cpw 4602  {csn 4628  {copab 5210  dom cdm 5676  ran crn 5677   ↾ cres 5678   “ cima 5679  Rel wrel 5681  Fun wfun 6535  Fincfn 8936

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 5299  ax-nul 5306  ax-pr 5427  ax-un 7722

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 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-om 7853  df-1o 8463  df-en 8937  df-fin 8940

This theorem is referenced by:  pwfi  9175