Theorem pwfiOLD 8892

Description: Obsolete version of pwfi 8776 as of 7-Sep-2024. (Contributed by NM, 26-Mar-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
pwfiOLD	⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)

Proof of Theorem pwfiOLD

Dummy variables 𝑚 𝑘 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfi 8579	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑚 ∈ ω 𝐴 ≈ 𝑚)
2		pweq 4504	. . . . . . 7 ⊢ (𝑚 = ∅ → 𝒫 𝑚 = 𝒫 ∅)
3	2	eleq1d 2817	. . . . . 6 ⊢ (𝑚 = ∅ → (𝒫 𝑚 ∈ Fin ↔ 𝒫 ∅ ∈ Fin))
4		pweq 4504	. . . . . . 7 ⊢ (𝑚 = 𝑘 → 𝒫 𝑚 = 𝒫 𝑘)
5	4	eleq1d 2817	. . . . . 6 ⊢ (𝑚 = 𝑘 → (𝒫 𝑚 ∈ Fin ↔ 𝒫 𝑘 ∈ Fin))
6		pweq 4504	. . . . . . . 8 ⊢ (𝑚 = suc 𝑘 → 𝒫 𝑚 = 𝒫 suc 𝑘)
7		df-suc 6178	. . . . . . . . 9 ⊢ suc 𝑘 = (𝑘 ∪ {𝑘})
8	7	pweqi 4506	. . . . . . . 8 ⊢ 𝒫 suc 𝑘 = 𝒫 (𝑘 ∪ {𝑘})
9	6, 8	eqtrdi 2789	. . . . . . 7 ⊢ (𝑚 = suc 𝑘 → 𝒫 𝑚 = 𝒫 (𝑘 ∪ {𝑘}))
10	9	eleq1d 2817	. . . . . 6 ⊢ (𝑚 = suc 𝑘 → (𝒫 𝑚 ∈ Fin ↔ 𝒫 (𝑘 ∪ {𝑘}) ∈ Fin))
11		pw0 4700	. . . . . . . 8 ⊢ 𝒫 ∅ = {∅}
12		df1o2 8143	. . . . . . . 8 ⊢ 1o = {∅}
13	11, 12	eqtr4i 2764	. . . . . . 7 ⊢ 𝒫 ∅ = 1o
14		1onn 8296	. . . . . . . 8 ⊢ 1o ∈ ω
15		ssid 3899	. . . . . . . 8 ⊢ 1o ⊆ 1o
16		ssnnfi 8768	. . . . . . . 8 ⊢ ((1o ∈ ω ∧ 1o ⊆ 1o) → 1o ∈ Fin)
17	14, 15, 16	mp2an 692	. . . . . . 7 ⊢ 1o ∈ Fin
18	13, 17	eqeltri 2829	. . . . . 6 ⊢ 𝒫 ∅ ∈ Fin
19		eqid 2738	. . . . . . . 8 ⊢ (𝑐 ∈ 𝒫 𝑘 ↦ (𝑐 ∪ {𝑘})) = (𝑐 ∈ 𝒫 𝑘 ↦ (𝑐 ∪ {𝑘}))
20	19	pwfilem 8775	. . . . . . 7 ⊢ (𝒫 𝑘 ∈ Fin → 𝒫 (𝑘 ∪ {𝑘}) ∈ Fin)
21	20	a1i 11	. . . . . 6 ⊢ (𝑘 ∈ ω → (𝒫 𝑘 ∈ Fin → 𝒫 (𝑘 ∪ {𝑘}) ∈ Fin))
22	3, 5, 10, 18, 21	finds1 7632	. . . . 5 ⊢ (𝑚 ∈ ω → 𝒫 𝑚 ∈ Fin)
23		pwen 8740	. . . . 5 ⊢ (𝐴 ≈ 𝑚 → 𝒫 𝐴 ≈ 𝒫 𝑚)
24		enfii 8784	. . . . 5 ⊢ ((𝒫 𝑚 ∈ Fin ∧ 𝒫 𝐴 ≈ 𝒫 𝑚) → 𝒫 𝐴 ∈ Fin)
25	22, 23, 24	syl2an 599	. . . 4 ⊢ ((𝑚 ∈ ω ∧ 𝐴 ≈ 𝑚) → 𝒫 𝐴 ∈ Fin)
26	25	rexlimiva 3191	. . 3 ⊢ (∃𝑚 ∈ ω 𝐴 ≈ 𝑚 → 𝒫 𝐴 ∈ Fin)
27	1, 26	sylbi 220	. 2 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin)
28		pwexr 7506	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ∈ V)
29		canth2g 8721	. . . 4 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴)
30		sdomdom 8583	. . . 4 ⊢ (𝐴 ≺ 𝒫 𝐴 → 𝐴 ≼ 𝒫 𝐴)
31	28, 29, 30	3syl 18	. . 3 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ≼ 𝒫 𝐴)
32		domfi 8817	. . 3 ⊢ ((𝒫 𝐴 ∈ Fin ∧ 𝐴 ≼ 𝒫 𝐴) → 𝐴 ∈ Fin)
33	31, 32	mpdan 687	. 2 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ∈ Fin)
34	27, 33	impbii 212	1 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1542   ∈ wcel 2114  ∃wrex 3054  Vcvv 3398   ∪ cun 3841   ⊆ wss 3843  ∅c0 4211  𝒫 cpw 4488  {csn 4516   class class class wbr 5030   ↦ cmpt 5110  suc csuc 6174  ωcom 7599  1_oc1o 8124   ≈ cen 8552   ≼ cdom 8553   ≺ csdm 8554  Fincfn 8555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-1o 8131  df-2o 8132  df-er 8320  df-map 8439  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559

This theorem is referenced by: (None)