Theorem pwfir 9195

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 5686	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵)
2		relopab 5821	. . . . 5 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
3		dmopabss 5916	. . . . 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 5934	. . 3 ⊢ ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
7		rnopab 5951	. . . 4 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
8		eleq1 2817	. . . . . . . . 9 ⊢ ({𝑦} = 𝑥 → ({𝑦} ∈ 𝒫 𝐵 ↔ 𝑥 ∈ 𝒫 𝐵))
9	8	biimparc 479	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → {𝑦} ∈ 𝒫 𝐵)
10		vex 3474	. . . . . . . . 9 ⊢ 𝑦 ∈ V
11	10	snelpw 5442	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 ↔ {𝑦} ∈ 𝒫 𝐵)
12	9, 11	sylibr 233	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → 𝑦 ∈ 𝐵)
13	12	exlimiv 1926	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → 𝑦 ∈ 𝐵)
14		snelpwi 5440	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ∈ 𝒫 𝐵)
15		eqid 2728	. . . . . . . 8 ⊢ {𝑦} = {𝑦}
16		eqeq2 2740	. . . . . . . . 9 ⊢ (𝑥 = {𝑦} → ({𝑦} = 𝑥 ↔ {𝑦} = {𝑦}))
17	16	rspcev 3608	. . . . . . . 8 ⊢ (({𝑦} ∈ 𝒫 𝐵 ∧ {𝑦} = {𝑦}) → ∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥)
18	14, 15, 17	sylancl 585	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥)
19		df-rex 3067	. . . . . . 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 2798	. . . 4 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = {𝑦 ∣ 𝑦 ∈ 𝐵}
23		abid2 2867	. . . 4 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐵} = 𝐵
24	7, 22, 23	3eqtri 2760	. . 3 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = 𝐵
25	1, 6, 24	3eqtri 2760	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) = 𝐵
26		funopab 6583	. . . 4 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥))
27		mosneq 4840	. . . . 5 ⊢ ∃*𝑦{𝑦} = 𝑥
28	27	moani 2543	. . . 4 ⊢ ∃*𝑦(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)
29	26, 28	mpgbir 1794	. . 3 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
30		imafi 9194	. . 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 2834	1 ⊢ (𝒫 𝐵 ∈ Fin → 𝐵 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ∃*wmo 2528  {cab 2705  ∃wrex 3066   ⊆ wss 3945  𝒫 cpw 4599  {csn 4625  {copab 5205  dom cdm 5673  ran crn 5674   ↾ cres 5675   “ cima 5676  Rel wrel 5678  Fun wfun 6537  Fincfn 8958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7866  df-1o 8481  df-en 8959  df-fin 8962

This theorem is referenced by:  pwfi  9197