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Theorem cff 8953
 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 8650 . 2 cf = (𝑥 ∈ On ↦ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))})
2 cardon 8653 . . . . . . 7 (card‘𝑧) ∈ On
3 eleq1 2676 . . . . . . 7 (𝑦 = (card‘𝑧) → (𝑦 ∈ On ↔ (card‘𝑧) ∈ On))
42, 3mpbiri 247 . . . . . 6 (𝑦 = (card‘𝑧) → 𝑦 ∈ On)
54adantr 480 . . . . 5 ((𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)) → 𝑦 ∈ On)
65exlimiv 1845 . . . 4 (∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)) → 𝑦 ∈ On)
76abssi 3640 . . 3 {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ⊆ On
8 cflem 8951 . . . 4 (𝑥 ∈ On → ∃𝑦𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)))
9 abn0 3908 . . . 4 ({𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ≠ ∅ ↔ ∃𝑦𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)))
108, 9sylibr 223 . . 3 (𝑥 ∈ On → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ≠ ∅)
11 oninton 6892 . . 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 694 . 2 (𝑥 ∈ On → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ∈ On)
131, 12fmpti 6291 1 cf:On⟶On
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410  Oncon0 5640  ⟶wf 5800  ‘cfv 5804  cardccrd 8644  cfccf 8646 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-card 8648  df-cf 8650 This theorem is referenced by:  cfub  8954  cardcf  8957  cflecard  8958  cfle  8959  cflim2  8968  cfidm  8980
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