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Mirrors > Home > MPE Home > Th. List > isf32lem12 | Structured version Visualization version GIF version |
Description: Lemma for isfin3-2 9589. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
isf32lem40.f | ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} |
Ref | Expression |
---|---|
isf32lem12 | ⊢ (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → 𝐺 ∈ 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8230 | . . . . 5 ⊢ (𝑓 ∈ (𝒫 𝐺 ↑𝑚 ω) → 𝑓:ω⟶𝒫 𝐺) | |
2 | isf32lem11 9585 | . . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) ∧ ¬ ∩ ran 𝑓 ∈ ran 𝑓)) → ω ≼* 𝐺) | |
3 | 2 | expcom 406 | . . . . . . . . 9 ⊢ ((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) ∧ ¬ ∩ ran 𝑓 ∈ ran 𝑓) → (𝐺 ∈ 𝑉 → ω ≼* 𝐺)) |
4 | 3 | 3expa 1098 | . . . . . . . 8 ⊢ (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏)) ∧ ¬ ∩ ran 𝑓 ∈ ran 𝑓) → (𝐺 ∈ 𝑉 → ω ≼* 𝐺)) |
5 | 4 | impancom 444 | . . . . . . 7 ⊢ (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏)) ∧ 𝐺 ∈ 𝑉) → (¬ ∩ ran 𝑓 ∈ ran 𝑓 → ω ≼* 𝐺)) |
6 | 5 | con1d 142 | . . . . . 6 ⊢ (((𝑓:ω⟶𝒫 𝐺 ∧ ∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏)) ∧ 𝐺 ∈ 𝑉) → (¬ ω ≼* 𝐺 → ∩ ran 𝑓 ∈ ran 𝑓)) |
7 | 6 | exp31 412 | . . . . 5 ⊢ (𝑓:ω⟶𝒫 𝐺 → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → ∩ ran 𝑓 ∈ ran 𝑓)))) |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝑓 ∈ (𝒫 𝐺 ↑𝑚 ω) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → ∩ ran 𝑓 ∈ ran 𝑓)))) |
9 | 8 | com4t 93 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → (𝑓 ∈ (𝒫 𝐺 ↑𝑚 ω) → (∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓)))) |
10 | 9 | ralrimdv 3138 | . 2 ⊢ (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → ∀𝑓 ∈ (𝒫 𝐺 ↑𝑚 ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓))) |
11 | isf32lem40.f | . . 3 ⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} | |
12 | 11 | isfin3ds 9551 | . 2 ⊢ (𝐺 ∈ 𝑉 → (𝐺 ∈ 𝐹 ↔ ∀𝑓 ∈ (𝒫 𝐺 ↑𝑚 ω)(∀𝑏 ∈ ω (𝑓‘suc 𝑏) ⊆ (𝑓‘𝑏) → ∩ ran 𝑓 ∈ ran 𝑓))) |
13 | 10, 12 | sylibrd 251 | 1 ⊢ (𝐺 ∈ 𝑉 → (¬ ω ≼* 𝐺 → 𝐺 ∈ 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 {cab 2758 ∀wral 3088 ⊆ wss 3831 𝒫 cpw 4423 ∩ cint 4750 class class class wbr 4930 ran crn 5409 suc csuc 6033 ⟶wf 6186 ‘cfv 6190 (class class class)co 6978 ωcom 7398 ↑𝑚 cmap 8208 ≼* cwdom 8818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-1o 7907 df-er 8091 df-map 8210 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-wdom 8820 df-card 9164 |
This theorem is referenced by: isf33lem 9588 isfin3-2 9589 |
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