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Theorem cff 10247
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 π‘₯ 𝑦 𝑧 𝑀 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cf 9940 . 2 cf = (π‘₯ ∈ On ↦ ∩ {𝑦 ∣ βˆƒπ‘§(𝑦 = (cardβ€˜π‘§) ∧ (𝑧 βŠ† π‘₯ ∧ βˆ€π‘€ ∈ π‘₯ βˆƒπ‘£ ∈ 𝑧 𝑀 βŠ† 𝑣))})
2 cardon 9943 . . . . . . 7 (cardβ€˜π‘§) ∈ On
3 eleq1 2820 . . . . . . 7 (𝑦 = (cardβ€˜π‘§) β†’ (𝑦 ∈ On ↔ (cardβ€˜π‘§) ∈ On))
42, 3mpbiri 258 . . . . . 6 (𝑦 = (cardβ€˜π‘§) β†’ 𝑦 ∈ On)
54adantr 480 . . . . 5 ((𝑦 = (cardβ€˜π‘§) ∧ (𝑧 βŠ† π‘₯ ∧ βˆ€π‘€ ∈ π‘₯ βˆƒπ‘£ ∈ 𝑧 𝑀 βŠ† 𝑣)) β†’ 𝑦 ∈ On)
65exlimiv 1932 . . . 4 (βˆƒπ‘§(𝑦 = (cardβ€˜π‘§) ∧ (𝑧 βŠ† π‘₯ ∧ βˆ€π‘€ ∈ π‘₯ βˆƒπ‘£ ∈ 𝑧 𝑀 βŠ† 𝑣)) β†’ 𝑦 ∈ On)
76abssi 4067 . . 3 {𝑦 ∣ βˆƒπ‘§(𝑦 = (cardβ€˜π‘§) ∧ (𝑧 βŠ† π‘₯ ∧ βˆ€π‘€ ∈ π‘₯ βˆƒπ‘£ ∈ 𝑧 𝑀 βŠ† 𝑣))} βŠ† On
8 cflem 10245 . . . 4 (π‘₯ ∈ On β†’ βˆƒπ‘¦βˆƒπ‘§(𝑦 = (cardβ€˜π‘§) ∧ (𝑧 βŠ† π‘₯ ∧ βˆ€π‘€ ∈ π‘₯ βˆƒπ‘£ ∈ 𝑧 𝑀 βŠ† 𝑣)))
9 abn0 4380 . . . 4 ({𝑦 ∣ βˆƒπ‘§(𝑦 = (cardβ€˜π‘§) ∧ (𝑧 βŠ† π‘₯ ∧ βˆ€π‘€ ∈ π‘₯ βˆƒπ‘£ ∈ 𝑧 𝑀 βŠ† 𝑣))} β‰  βˆ… ↔ βˆƒπ‘¦βˆƒπ‘§(𝑦 = (cardβ€˜π‘§) ∧ (𝑧 βŠ† π‘₯ ∧ βˆ€π‘€ ∈ π‘₯ βˆƒπ‘£ ∈ 𝑧 𝑀 βŠ† 𝑣)))
108, 9sylibr 233 . . 3 (π‘₯ ∈ On β†’ {𝑦 ∣ βˆƒπ‘§(𝑦 = (cardβ€˜π‘§) ∧ (𝑧 βŠ† π‘₯ ∧ βˆ€π‘€ ∈ π‘₯ βˆƒπ‘£ ∈ 𝑧 𝑀 βŠ† 𝑣))} β‰  βˆ…)
11 oninton 7787 . . 3 (({𝑦 ∣ βˆƒπ‘§(𝑦 = (cardβ€˜π‘§) ∧ (𝑧 βŠ† π‘₯ ∧ βˆ€π‘€ ∈ π‘₯ βˆƒπ‘£ ∈ 𝑧 𝑀 βŠ† 𝑣))} βŠ† On ∧ {𝑦 ∣ βˆƒπ‘§(𝑦 = (cardβ€˜π‘§) ∧ (𝑧 βŠ† π‘₯ ∧ βˆ€π‘€ ∈ π‘₯ βˆƒπ‘£ ∈ 𝑧 𝑀 βŠ† 𝑣))} β‰  βˆ…) β†’ ∩ {𝑦 ∣ βˆƒπ‘§(𝑦 = (cardβ€˜π‘§) ∧ (𝑧 βŠ† π‘₯ ∧ βˆ€π‘€ ∈ π‘₯ βˆƒπ‘£ ∈ 𝑧 𝑀 βŠ† 𝑣))} ∈ On)
127, 10, 11sylancr 586 . 2 (π‘₯ ∈ On β†’ ∩ {𝑦 ∣ βˆƒπ‘§(𝑦 = (cardβ€˜π‘§) ∧ (𝑧 βŠ† π‘₯ ∧ βˆ€π‘€ ∈ π‘₯ βˆƒπ‘£ ∈ 𝑧 𝑀 βŠ† 𝑣))} ∈ On)
131, 12fmpti 7113 1 cf:On⟢On
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 395   = wceq 1540  βˆƒwex 1780   ∈ wcel 2105  {cab 2708   β‰  wne 2939  βˆ€wral 3060  βˆƒwrex 3069   βŠ† wss 3948  βˆ…c0 4322  βˆ© cint 4950  Oncon0 6364  βŸΆwf 6539  β€˜cfv 6543  cardccrd 9934  cfccf 9936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-card 9938  df-cf 9940
This theorem is referenced by:  cfub  10248  cardcf  10251  cflecard  10252  cfle  10253  cflim2  10262  cfidm  10274
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