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Mirrors > Home > MPE Home > Th. List > pwfiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of pwfi 8833 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 8630 | . . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑚 ∈ ω 𝐴 ≈ 𝑚) | |
2 | pweq 4515 | . . . . . . 7 ⊢ (𝑚 = ∅ → 𝒫 𝑚 = 𝒫 ∅) | |
3 | 2 | eleq1d 2815 | . . . . . 6 ⊢ (𝑚 = ∅ → (𝒫 𝑚 ∈ Fin ↔ 𝒫 ∅ ∈ Fin)) |
4 | pweq 4515 | . . . . . . 7 ⊢ (𝑚 = 𝑘 → 𝒫 𝑚 = 𝒫 𝑘) | |
5 | 4 | eleq1d 2815 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝒫 𝑚 ∈ Fin ↔ 𝒫 𝑘 ∈ Fin)) |
6 | pweq 4515 | . . . . . . . 8 ⊢ (𝑚 = suc 𝑘 → 𝒫 𝑚 = 𝒫 suc 𝑘) | |
7 | df-suc 6197 | . . . . . . . . 9 ⊢ suc 𝑘 = (𝑘 ∪ {𝑘}) | |
8 | 7 | pweqi 4517 | . . . . . . . 8 ⊢ 𝒫 suc 𝑘 = 𝒫 (𝑘 ∪ {𝑘}) |
9 | 6, 8 | eqtrdi 2787 | . . . . . . 7 ⊢ (𝑚 = suc 𝑘 → 𝒫 𝑚 = 𝒫 (𝑘 ∪ {𝑘})) |
10 | 9 | eleq1d 2815 | . . . . . 6 ⊢ (𝑚 = suc 𝑘 → (𝒫 𝑚 ∈ Fin ↔ 𝒫 (𝑘 ∪ {𝑘}) ∈ Fin)) |
11 | pw0 4711 | . . . . . . . 8 ⊢ 𝒫 ∅ = {∅} | |
12 | df1o2 8192 | . . . . . . . 8 ⊢ 1o = {∅} | |
13 | 11, 12 | eqtr4i 2762 | . . . . . . 7 ⊢ 𝒫 ∅ = 1o |
14 | 1onn 8345 | . . . . . . . 8 ⊢ 1o ∈ ω | |
15 | ssid 3909 | . . . . . . . 8 ⊢ 1o ⊆ 1o | |
16 | ssnnfi 8825 | . . . . . . . 8 ⊢ ((1o ∈ ω ∧ 1o ⊆ 1o) → 1o ∈ Fin) | |
17 | 14, 15, 16 | mp2an 692 | . . . . . . 7 ⊢ 1o ∈ Fin |
18 | 13, 17 | eqeltri 2827 | . . . . . 6 ⊢ 𝒫 ∅ ∈ Fin |
19 | eqid 2736 | . . . . . . . 8 ⊢ (𝑐 ∈ 𝒫 𝑘 ↦ (𝑐 ∪ {𝑘})) = (𝑐 ∈ 𝒫 𝑘 ↦ (𝑐 ∪ {𝑘})) | |
20 | 19 | pwfilem 8832 | . . . . . . 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 7657 | . . . . 5 ⊢ (𝑚 ∈ ω → 𝒫 𝑚 ∈ Fin) |
23 | pwen 8797 | . . . . 5 ⊢ (𝐴 ≈ 𝑚 → 𝒫 𝐴 ≈ 𝒫 𝑚) | |
24 | enfii 8841 | . . . . 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 3190 | . . 3 ⊢ (∃𝑚 ∈ ω 𝐴 ≈ 𝑚 → 𝒫 𝐴 ∈ Fin) |
27 | 1, 26 | sylbi 220 | . 2 ⊢ (𝐴 ∈ Fin → 𝒫 𝐴 ∈ Fin) |
28 | pwexr 7528 | . . . 4 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ∈ V) | |
29 | canth2g 8778 | . . . 4 ⊢ (𝐴 ∈ V → 𝐴 ≺ 𝒫 𝐴) | |
30 | sdomdom 8634 | . . . 4 ⊢ (𝐴 ≺ 𝒫 𝐴 → 𝐴 ≼ 𝒫 𝐴) | |
31 | 28, 29, 30 | 3syl 18 | . . 3 ⊢ (𝒫 𝐴 ∈ Fin → 𝐴 ≼ 𝒫 𝐴) |
32 | domfi 8874 | . . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 Vcvv 3398 ∪ cun 3851 ⊆ wss 3853 ∅c0 4223 𝒫 cpw 4499 {csn 4527 class class class wbr 5039 ↦ cmpt 5120 suc csuc 6193 ωcom 7622 1oc1o 8173 ≈ cen 8601 ≼ cdom 8602 ≺ csdm 8603 Fincfn 8604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-1o 8180 df-2o 8181 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 |
This theorem is referenced by: (None) |
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