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Theorem cff 9658
Description: Cofinality is a function on the class of ordinal numbers to the class of cardinal numbers. (Contributed by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cff cf:On⟶On

Proof of Theorem cff
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cf 9358 . 2 cf = (𝑥 ∈ On ↦ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))})
2 cardon 9361 . . . . . . 7 (card‘𝑧) ∈ On
3 eleq1 2897 . . . . . . 7 (𝑦 = (card‘𝑧) → (𝑦 ∈ On ↔ (card‘𝑧) ∈ On))
42, 3mpbiri 259 . . . . . 6 (𝑦 = (card‘𝑧) → 𝑦 ∈ On)
54adantr 481 . . . . 5 ((𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)) → 𝑦 ∈ On)
65exlimiv 1922 . . . 4 (∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)) → 𝑦 ∈ On)
76abssi 4043 . . 3 {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ⊆ On
8 cflem 9656 . . . 4 (𝑥 ∈ On → ∃𝑦𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)))
9 abn0 4333 . . . 4 ({𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ≠ ∅ ↔ ∃𝑦𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)))
108, 9sylibr 235 . . 3 (𝑥 ∈ On → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ≠ ∅)
11 oninton 7504 . . 3 (({𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ⊆ On ∧ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ≠ ∅) → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ∈ On)
127, 10, 11sylancr 587 . 2 (𝑥 ∈ On → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ∈ On)
131, 12fmpti 6868 1 cf:On⟶On
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wne 3013  wral 3135  wrex 3136  wss 3933  c0 4288   cint 4867  Oncon0 6184  wf 6344  cfv 6348  cardccrd 9352  cfccf 9354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-card 9356  df-cf 9358
This theorem is referenced by:  cfub  9659  cardcf  9662  cflecard  9663  cfle  9664  cflim2  9673  cfidm  9685
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