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Theorem cfval 10238
Description: Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number 𝐴 is the cardinality (size) of the smallest unbounded subset 𝑦 of the ordinal number. Unbounded means that for every member of 𝐴, there is a member of 𝑦 that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is. (Contributed by NM, 1-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cfval (𝐴 ∈ On β†’ (cfβ€˜π΄) = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
Distinct variable group:   π‘₯,𝑦,𝑧,𝑀,𝐴

Proof of Theorem cfval
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 cflem 10237 . . 3 (𝐴 ∈ On β†’ βˆƒπ‘₯βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
2 intexab 5338 . . 3 (βˆƒπ‘₯βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} ∈ V)
31, 2sylib 217 . 2 (𝐴 ∈ On β†’ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} ∈ V)
4 sseq2 4007 . . . . . . . 8 (𝑣 = 𝐴 β†’ (𝑦 βŠ† 𝑣 ↔ 𝑦 βŠ† 𝐴))
5 raleq 3322 . . . . . . . 8 (𝑣 = 𝐴 β†’ (βˆ€π‘§ ∈ 𝑣 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))
64, 5anbi12d 631 . . . . . . 7 (𝑣 = 𝐴 β†’ ((𝑦 βŠ† 𝑣 ∧ βˆ€π‘§ ∈ 𝑣 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀) ↔ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
76anbi2d 629 . . . . . 6 (𝑣 = 𝐴 β†’ ((π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝑣 ∧ βˆ€π‘§ ∈ 𝑣 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ (π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
87exbidv 1924 . . . . 5 (𝑣 = 𝐴 β†’ (βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝑣 ∧ βˆ€π‘§ ∈ 𝑣 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)) ↔ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))))
98abbidv 2801 . . . 4 (𝑣 = 𝐴 β†’ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝑣 ∧ βˆ€π‘§ ∈ 𝑣 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} = {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
109inteqd 4954 . . 3 (𝑣 = 𝐴 β†’ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝑣 ∧ βˆ€π‘§ ∈ 𝑣 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
11 df-cf 9932 . . 3 cf = (𝑣 ∈ On ↦ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝑣 ∧ βˆ€π‘§ ∈ 𝑣 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
1210, 11fvmptg 6993 . 2 ((𝐴 ∈ On ∧ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))} ∈ V) β†’ (cfβ€˜π΄) = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
133, 12mpdan 685 1 (𝐴 ∈ On β†’ (cfβ€˜π΄) = ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3947  βˆ© cint 4949  Oncon0 6361  β€˜cfv 6540  cardccrd 9926  cfccf 9928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-cf 9932
This theorem is referenced by:  cfub  10240  cflm  10241  cardcf  10243  cflecard  10244  cfeq0  10247  cfsuc  10248  cff1  10249  cfflb  10250  cfval2  10251  cflim3  10253
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