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Mirrors > Home > MPE Home > Th. List > isf33lem | Structured version Visualization version GIF version |
Description: Lemma for isfin3-3 9779. (Contributed by Stefan O'Rear, 17-May-2015.) |
Ref | Expression |
---|---|
isf33lem | ⊢ FinIII = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfin32i 9776 | . . . 4 ⊢ (𝑓 ∈ FinIII → ¬ ω ≼* 𝑓) | |
2 | fveq1 6644 | . . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎‘suc 𝑥) = (𝑏‘suc 𝑥)) | |
3 | fveq1 6644 | . . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎‘𝑥) = (𝑏‘𝑥)) | |
4 | 2, 3 | sseq12d 3948 | . . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥))) |
5 | 4 | ralbidv 3162 | . . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) ↔ ∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥))) |
6 | rneq 5770 | . . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → ran 𝑎 = ran 𝑏) | |
7 | 6 | inteqd 4843 | . . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ∩ ran 𝑎 = ∩ ran 𝑏) |
8 | 7, 6 | eleq12d 2884 | . . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (∩ ran 𝑎 ∈ ran 𝑎 ↔ ∩ ran 𝑏 ∈ ran 𝑏)) |
9 | 5, 8 | imbi12d 348 | . . . . . . . 8 ⊢ (𝑎 = 𝑏 → ((∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎) ↔ (∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏))) |
10 | 9 | cbvralvw 3396 | . . . . . . 7 ⊢ (∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎) ↔ ∀𝑏 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏)) |
11 | pweq 4513 | . . . . . . . . 9 ⊢ (𝑔 = 𝑦 → 𝒫 𝑔 = 𝒫 𝑦) | |
12 | 11 | oveq1d 7150 | . . . . . . . 8 ⊢ (𝑔 = 𝑦 → (𝒫 𝑔 ↑m ω) = (𝒫 𝑦 ↑m ω)) |
13 | 12 | raleqdv 3364 | . . . . . . 7 ⊢ (𝑔 = 𝑦 → (∀𝑏 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏) ↔ ∀𝑏 ∈ (𝒫 𝑦 ↑m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏))) |
14 | 10, 13 | syl5bb 286 | . . . . . 6 ⊢ (𝑔 = 𝑦 → (∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎) ↔ ∀𝑏 ∈ (𝒫 𝑦 ↑m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏))) |
15 | 14 | cbvabv 2866 | . . . . 5 ⊢ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} = {𝑦 ∣ ∀𝑏 ∈ (𝒫 𝑦 ↑m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏)} |
16 | 15 | isf32lem12 9775 | . . . 4 ⊢ (𝑓 ∈ FinIII → (¬ ω ≼* 𝑓 → 𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)})) |
17 | 1, 16 | mpd 15 | . . 3 ⊢ (𝑓 ∈ FinIII → 𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}) |
18 | 10 | abbii 2863 | . . . 4 ⊢ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} = {𝑔 ∣ ∀𝑏 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏‘𝑥) → ∩ ran 𝑏 ∈ ran 𝑏)} |
19 | 18 | fin23lem41 9763 | . . 3 ⊢ (𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} → 𝑓 ∈ FinIII) |
20 | 17, 19 | impbii 212 | . 2 ⊢ (𝑓 ∈ FinIII ↔ 𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}) |
21 | 20 | eqriv 2795 | 1 ⊢ FinIII = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 {cab 2776 ∀wral 3106 ⊆ wss 3881 𝒫 cpw 4497 ∩ cint 4838 class class class wbr 5030 ran crn 5520 suc csuc 6161 ‘cfv 6324 (class class class)co 7135 ωcom 7560 ↑m cmap 8389 ≼* cwdom 9012 FinIIIcfin3 9692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-seqom 8067 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-wdom 9013 df-card 9352 df-fin4 9698 df-fin3 9699 |
This theorem is referenced by: isfin3-2 9778 isfin3-3 9779 fin23 9800 |
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