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Mirrors > Home > MPE Home > Th. List > pwfiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of pwfi 9081 as of 7-Sep-2024. (Contributed by NM, 26-Mar-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pwfiOLD | ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 8875 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑚 ∈ ω 𝐴 ≈ 𝑚) | |
2 | pweq 4573 | . . . . . . 7 ⊢ (𝑚 = ∅ → 𝒫 𝑚 = 𝒫 ∅) | |
3 | 2 | eleq1d 2823 | . . . . . 6 ⊢ (𝑚 = ∅ → (𝒫 𝑚 ∈ Fin ↔ 𝒫 ∅ ∈ Fin)) |
4 | pweq 4573 | . . . . . . 7 ⊢ (𝑚 = 𝑘 → 𝒫 𝑚 = 𝒫 𝑘) | |
5 | 4 | eleq1d 2823 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝒫 𝑚 ∈ Fin ↔ 𝒫 𝑘 ∈ Fin)) |
6 | pweq 4573 | . . . . . . . 8 ⊢ (𝑚 = suc 𝑘 → 𝒫 𝑚 = 𝒫 suc 𝑘) | |
7 | df-suc 6322 | . . . . . . . . 9 ⊢ suc 𝑘 = (𝑘 ∪ {𝑘}) | |
8 | 7 | pweqi 4575 | . . . . . . . 8 ⊢ 𝒫 suc 𝑘 = 𝒫 (𝑘 ∪ {𝑘}) |
9 | 6, 8 | eqtrdi 2794 | . . . . . . 7 ⊢ (𝑚 = suc 𝑘 → 𝒫 𝑚 = 𝒫 (𝑘 ∪ {𝑘})) |
10 | 9 | eleq1d 2823 | . . . . . 6 ⊢ (𝑚 = suc 𝑘 → (𝒫 𝑚 ∈ Fin ↔ 𝒫 (𝑘 ∪ {𝑘}) ∈ Fin)) |
11 | pw0 4771 | . . . . . . . 8 ⊢ 𝒫 ∅ = {∅} | |
12 | df1o2 8412 | . . . . . . . 8 ⊢ 1o = {∅} | |
13 | 11, 12 | eqtr4i 2769 | . . . . . . 7 ⊢ 𝒫 ∅ = 1o |
14 | 1onn 8579 | . . . . . . . 8 ⊢ 1o ∈ ω | |
15 | ssid 3965 | . . . . . . . 8 ⊢ 1o ⊆ 1o | |
16 | ssnnfi 9072 | . . . . . . . 8 ⊢ ((1o ∈ ω ∧ 1o ⊆ 1o) → 1o ∈ Fin) | |
17 | 14, 15, 16 | mp2an 691 | . . . . . . 7 ⊢ 1o ∈ Fin |
18 | 13, 17 | eqeltri 2835 | . . . . . 6 ⊢ 𝒫 ∅ ∈ Fin |
19 | eqid 2738 | . . . . . . . 8 ⊢ (𝑐 ∈ 𝒫 𝑘 ↦ (𝑐 ∪ {𝑘})) = (𝑐 ∈ 𝒫 𝑘 ↦ (𝑐 ∪ {𝑘})) | |
20 | 19 | pwfilem 9080 | . . . . . . 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 7831 | . . . . 5 ⊢ (𝑚 ∈ ω → 𝒫 𝑚 ∈ Fin) |
23 | pwen 9053 | . . . . 5 ⊢ (𝐴 ≈ 𝑚 → 𝒫 𝐴 ≈ 𝒫 𝑚) | |
24 | enfii 9092 | . . . . 5 ⊢ ((𝒫 𝑚 ∈ Fin ∧ 𝒫 𝐴 ≈ 𝒫 𝑚) → 𝒫 𝐴 ∈ Fin) | |
25 | 22, 23, 24 | syl2an 597 | . . . 4 ⊢ ((𝑚 ∈ ω ∧ 𝐴 ≈ 𝑚) → 𝒫 𝐴 ∈ Fin) |
26 | 25 | rexlimiva 3143 | . . 3 ⊢ (∃𝑚 ∈ ω 𝐴 ≈ 𝑚 → 𝒫 𝐴 ∈ Fin) |
27 | 1, 26 | sylbi 216 | . 2 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin) |
28 | pwexr 7692 | . . . 4 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ∈ V) | |
29 | canth2g 9034 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴) | |
30 | sdomdom 8879 | . . . 4 ⊢ (𝐴 ≺ 𝒫 𝐴 → 𝐴 ≼ 𝒫 𝐴) | |
31 | 28, 29, 30 | 3syl 18 | . . 3 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ≼ 𝒫 𝐴) |
32 | domfi 9095 | . . 3 ⊢ ((𝒫 𝐴 ∈ Fin ∧ 𝐴 ≼ 𝒫 𝐴) → 𝐴 ∈ Fin) | |
33 | 31, 32 | mpdan 686 | . 2 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ∈ Fin) |
34 | 27, 33 | impbii 208 | 1 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∃wrex 3072 Vcvv 3444 ∪ cun 3907 ⊆ wss 3909 ∅c0 4281 𝒫 cpw 4559 {csn 4585 class class class wbr 5104 ↦ cmpt 5187 suc csuc 6318 ωcom 7795 1oc1o 8398 ≈ cen 8839 ≼ cdom 8840 ≺ csdm 8841 Fincfn 8842 |
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 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 |
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 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-1o 8405 df-2o 8406 df-er 8607 df-map 8726 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 |
This theorem is referenced by: (None) |
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