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Theorem cfval 9657
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 → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝐴

Proof of Theorem cfval
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 cflem 9656 . . 3 (𝐴 ∈ On → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
2 intexab 5233 . . 3 (∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ V)
31, 2sylib 219 . 2 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ V)
4 sseq2 3990 . . . . . . . 8 (𝑣 = 𝐴 → (𝑦𝑣𝑦𝐴))
5 raleq 3403 . . . . . . . 8 (𝑣 = 𝐴 → (∀𝑧𝑣𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
64, 5anbi12d 630 . . . . . . 7 (𝑣 = 𝐴 → ((𝑦𝑣 ∧ ∀𝑧𝑣𝑤𝑦 𝑧𝑤) ↔ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
76anbi2d 628 . . . . . 6 (𝑣 = 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝑣 ∧ ∀𝑧𝑣𝑤𝑦 𝑧𝑤)) ↔ (𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
87exbidv 1913 . . . . 5 (𝑣 = 𝐴 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝑣 ∧ ∀𝑧𝑣𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
98abbidv 2882 . . . 4 (𝑣 = 𝐴 → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝑣 ∧ ∀𝑧𝑣𝑤𝑦 𝑧𝑤))} = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
109inteqd 4872 . . 3 (𝑣 = 𝐴 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝑣 ∧ ∀𝑧𝑣𝑤𝑦 𝑧𝑤))} = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
11 df-cf 9358 . . 3 cf = (𝑣 ∈ On ↦ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝑣 ∧ ∀𝑧𝑣𝑤𝑦 𝑧𝑤))})
1210, 11fvmptg 6759 . 2 ((𝐴 ∈ On ∧ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ V) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
133, 12mpdan 683 1 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  {cab 2796  wral 3135  wrex 3136  Vcvv 3492  wss 3933   cint 4867  Oncon0 6184  cfv 6348  cardccrd 9352  cfccf 9354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-cf 9358
This theorem is referenced by:  cfub  9659  cflm  9660  cardcf  9662  cflecard  9663  cfeq0  9666  cfsuc  9667  cff1  9668  cfflb  9669  cfval2  9670  cflim3  9672
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