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Theorem cfval 10188
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 10187 . . 3 (𝐴 ∈ On β†’ βˆƒπ‘₯βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝐴 ∧ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀)))
2 intexab 5297 . . 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 3971 . . . . . . . 8 (𝑣 = 𝐴 β†’ (𝑦 βŠ† 𝑣 ↔ 𝑦 βŠ† 𝐴))
5 raleq 3308 . . . . . . . 8 (𝑣 = 𝐴 β†’ (βˆ€π‘§ ∈ 𝑣 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀 ↔ βˆ€π‘§ ∈ 𝐴 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))
64, 5anbi12d 632 . . . . . . 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 630 . . . . . 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 1925 . . . . 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 2802 . . . 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 4913 . . 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 9882 . . 3 cf = (𝑣 ∈ On ↦ ∩ {π‘₯ ∣ βˆƒπ‘¦(π‘₯ = (cardβ€˜π‘¦) ∧ (𝑦 βŠ† 𝑣 ∧ βˆ€π‘§ ∈ 𝑣 βˆƒπ‘€ ∈ 𝑦 𝑧 βŠ† 𝑀))})
1210, 11fvmptg 6947 . 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 686 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 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3444   βŠ† wss 3911  βˆ© cint 4908  Oncon0 6318  β€˜cfv 6497  cardccrd 9876  cfccf 9878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-cf 9882
This theorem is referenced by:  cfub  10190  cflm  10191  cardcf  10193  cflecard  10194  cfeq0  10197  cfsuc  10198  cff1  10199  cfflb  10200  cfval2  10201  cflim3  10203
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