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Theorem cff 9282
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 8977 . 2 cf = (𝑥 ∈ On ↦ {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))})
2 cardon 8980 . . . . . . 7 (card‘𝑧) ∈ On
3 eleq1 2827 . . . . . . 7 (𝑦 = (card‘𝑧) → (𝑦 ∈ On ↔ (card‘𝑧) ∈ On))
42, 3mpbiri 248 . . . . . 6 (𝑦 = (card‘𝑧) → 𝑦 ∈ On)
54adantr 472 . . . . 5 ((𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)) → 𝑦 ∈ On)
65exlimiv 2007 . . . 4 (∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)) → 𝑦 ∈ On)
76abssi 3818 . . 3 {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ⊆ On
8 cflem 9280 . . . 4 (𝑥 ∈ On → ∃𝑦𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)))
9 abn0 4097 . . . 4 ({𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ≠ ∅ ↔ ∃𝑦𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣)))
108, 9sylibr 224 . . 3 (𝑥 ∈ On → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ≠ ∅)
11 oninton 7166 . . 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 698 . 2 (𝑥 ∈ On → {𝑦 ∣ ∃𝑧(𝑦 = (card‘𝑧) ∧ (𝑧𝑥 ∧ ∀𝑤𝑥𝑣𝑧 𝑤𝑣))} ∈ On)
131, 12fmpti 6547 1 cf:On⟶On
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wne 2932  wral 3050  wrex 3051  wss 3715  c0 4058   cint 4627  Oncon0 5884  wf 6045  cfv 6049  cardccrd 8971  cfccf 8973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-card 8975  df-cf 8977
This theorem is referenced by:  cfub  9283  cardcf  9286  cflecard  9287  cfle  9288  cflim2  9297  cfidm  9309
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