Theorem pwfir 9255

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 5656	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) = ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵)
2		relopab 5793	. . . . 5 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
3		dmopabss 5890	. . . . 5 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ⊆ 𝒫 𝐵
4		relssres 6004	. . . . 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 702	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
6	5	rneqi 5909	. . 3 ⊢ ran ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↾ 𝒫 𝐵) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
7		rnopab 5926	. . . 4 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
8		eleq1 2849	. . . . . . . . 9 ⊢ ({𝑦} = 𝑥 → ({𝑦} ∈ 𝒫 𝐵 ↔ 𝑥 ∈ 𝒫 𝐵))
9	8	biimparc 483	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → {𝑦} ∈ 𝒫 𝐵)
10		vex 3457	. . . . . . . . 9 ⊢ 𝑦 ∈ V
11	10	snelpw 5409	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 ↔ {𝑦} ∈ 𝒫 𝐵)
12	9, 11	sylibr 236	. . . . . . 7 ⊢ ((𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → 𝑦 ∈ 𝐵)
13	12	exlimiv 1949	. . . . . 6 ⊢ (∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) → 𝑦 ∈ 𝐵)
14		snelpwi 5408	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → {𝑦} ∈ 𝒫 𝐵)
15		eqid 2761	. . . . . . . 8 ⊢ {𝑦} = {𝑦}
16		eqeq2 2773	. . . . . . . . 9 ⊢ (𝑥 = {𝑦} → ({𝑦} = 𝑥 ↔ {𝑦} = {𝑦}))
17	16	rspcev 3580	. . . . . . . 8 ⊢ (({𝑦} ∈ 𝒫 𝐵 ∧ {𝑦} = {𝑦}) → ∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥)
18	14, 15, 17	sylancl 595	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥)
19		df-rex 3086	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝒫 𝐵{𝑦} = 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥))
20	18, 19	sylib 220	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥))
21	13, 20	impbii 211	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥) ↔ 𝑦 ∈ 𝐵)
22	21	abbii 2828	. . . 4 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = {𝑦 ∣ 𝑦 ∈ 𝐵}
23		abid2 2898	. . . 4 ⊢ {𝑦 ∣ 𝑦 ∈ 𝐵} = 𝐵
24	7, 22, 23	3eqtri 2788	. . 3 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} = 𝐵
25	1, 6, 24	3eqtri 2788	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) = 𝐵
26		funopab 6551	. . . 4 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥))
27		mosneq 4797	. . . . 5 ⊢ ∃*𝑦{𝑦} = 𝑥
28	27	moani 2579	. . . 4 ⊢ ∃*𝑦(𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)
29	26, 28	mpgbir 1818	. . 3 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)}
30		imafi 9253	. . 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 700	. 2 ⊢ (𝒫 𝐵 ∈ Fin → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝒫 𝐵 ∧ {𝑦} = 𝑥)} “ 𝒫 𝐵) ∈ Fin)
32	25, 31	eqeltrrid 2866	1 ⊢ (𝒫 𝐵 ∈ Fin → 𝐵 ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∃*wmo 2563  {cab 2739  ∃wrex 3085   ⊆ wss 3902  𝒫 cpw 4552  {csn 4579  {copab 5159  dom cdm 5643  ran crn 5644   ↾ cres 5645   “ cima 5646  Rel wrel 5648  Fun wfun 6510  Fincfn 8921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-om 7842  df-1o 8431  df-en 8922  df-dom 8923  df-fin 8925

This theorem is referenced by:  pwfi  9257  r1omfi  35362