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Mirrors > Home > MPE Home > Th. List > cff | Structured version Visualization version GIF version |
Description: Cofinality is a function on the class of ordinal numbers to the class of cardinal numbers. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
cff | ⊢ cf:On⟶On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cf 9372 | . 2 ⊢ cf = (𝑥 ∈ On ↦ ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))}) | |
2 | cardon 9375 | . . . . . . 7 ⊢ (card‘𝑧) ∈ On | |
3 | eleq1 2902 | . . . . . . 7 ⊢ (𝑦 = (card‘𝑧) → (𝑦 ∈ On ↔ (card‘𝑧) ∈ On)) | |
4 | 2, 3 | mpbiri 260 | . . . . . 6 ⊢ (𝑦 = (card‘𝑧) → 𝑦 ∈ On) |
5 | 4 | adantr 483 | . . . . 5 ⊢ ((𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣)) → 𝑦 ∈ On) |
6 | 5 | exlimiv 1931 | . . . 4 ⊢ (∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣)) → 𝑦 ∈ On) |
7 | 6 | abssi 4048 | . . 3 ⊢ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ⊆ On |
8 | cflem 9670 | . . . 4 ⊢ (𝑥 ∈ On → ∃𝑦∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))) | |
9 | abn0 4338 | . . . 4 ⊢ ({𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ≠ ∅ ↔ ∃𝑦∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))) | |
10 | 8, 9 | sylibr 236 | . . 3 ⊢ (𝑥 ∈ On → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ≠ ∅) |
11 | oninton 7517 | . . 3 ⊢ (({𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ⊆ On ∧ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ≠ ∅) → ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ∈ On) | |
12 | 7, 10, 11 | sylancr 589 | . 2 ⊢ (𝑥 ∈ On → ∩ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧 ⊆ 𝑥 ∧ ∀𝑤 ∈ 𝑥 ∃𝑣 ∈ 𝑧 𝑤 ⊆ 𝑣))} ∈ On) |
13 | 1, 12 | fmpti 6878 | 1 ⊢ cf:On⟶On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2801 ≠ wne 3018 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 ∅c0 4293 ∩ cint 4878 Oncon0 6193 ⟶wf 6353 ‘cfv 6357 cardccrd 9366 cfccf 9368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-card 9370 df-cf 9372 |
This theorem is referenced by: cfub 9673 cardcf 9676 cflecard 9677 cfle 9678 cflim2 9687 cfidm 9699 |
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