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Theorem cff 9355
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 9050 . 2 cf = (𝑥 ∈ On ↦ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))})
2 cardon 9053 . . . . . . 7 (card‘𝑧) ∈ On
3 eleq1 2873 . . . . . . 7 (𝑦 = (card‘𝑧) → (𝑦 ∈ On ↔ (card‘𝑧) ∈ On))
42, 3mpbiri 249 . . . . . 6 (𝑦 = (card‘𝑧) → 𝑦 ∈ On)
54adantr 468 . . . . 5 ((𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)) → 𝑦 ∈ On)
65exlimiv 2021 . . . 4 (∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)) → 𝑦 ∈ On)
76abssi 3874 . . 3 {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ⊆ On
8 cflem 9353 . . . 4 (𝑥 ∈ On → ∃𝑦𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)))
9 abn0 4155 . . . 4 ({𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ≠ ∅ ↔ ∃𝑦𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)))
108, 9sylibr 225 . . 3 (𝑥 ∈ On → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ≠ ∅)
11 oninton 7230 . . 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 577 . 2 (𝑥 ∈ On → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ∈ On)
131, 12fmpti 6604 1 cf:On⟶On
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  wex 1859  wcel 2156  {cab 2792  wne 2978  wral 3096  wrex 3097  wss 3769  c0 4116   cint 4669  Oncon0 5936  wf 6097  cfv 6101  cardccrd 9044  cfccf 9046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-ord 5939  df-on 5940  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-card 9048  df-cf 9050
This theorem is referenced by:  cfub  9356  cardcf  9359  cflecard  9360  cfle  9361  cflim2  9370  cfidm  9382
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