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Theorem cff 8931
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 8628 . 2 cf = (𝑥 ∈ On ↦ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))})
2 cardon 8631 . . . . . . 7 (card‘𝑧) ∈ On
3 eleq1 2675 . . . . . . 7 (𝑦 = (card‘𝑧) → (𝑦 ∈ On ↔ (card‘𝑧) ∈ On))
42, 3mpbiri 246 . . . . . 6 (𝑦 = (card‘𝑧) → 𝑦 ∈ On)
54adantr 479 . . . . 5 ((𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)) → 𝑦 ∈ On)
65exlimiv 1844 . . . 4 (∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)) → 𝑦 ∈ On)
76abssi 3639 . . 3 {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ⊆ On
8 cflem 8929 . . . 4 (𝑥 ∈ On → ∃𝑦𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)))
9 abn0 3907 . . . 4 ({𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ≠ ∅ ↔ ∃𝑦𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)))
108, 9sylibr 222 . . 3 (𝑥 ∈ On → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ≠ ∅)
11 oninton 6870 . . 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 693 . 2 (𝑥 ∈ On → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ∈ On)
131, 12fmpti 6276 1 cf:On⟶On
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wex 1694  wcel 1976  {cab 2595  wne 2779  wral 2895  wrex 2896  wss 3539  c0 3873   cint 4404  Oncon0 5626  wf 5786  cfv 5790  cardccrd 8622  cfccf 8624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-card 8626  df-cf 8628
This theorem is referenced by:  cfub  8932  cardcf  8935  cflecard  8936  cfle  8937  cflim2  8946  cfidm  8958
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