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Theorem cfval 9270
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 9269 . . 3 (𝐴 ∈ On → ∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
2 intexab 4950 . . 3 (∃𝑥𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)) ↔ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ V)
31, 2sylib 208 . 2 (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ V)
4 sseq2 3774 . . . . . . . 8 (𝑣 = 𝐴 → (𝑦𝑣𝑦𝐴))
5 raleq 3286 . . . . . . . 8 (𝑣 = 𝐴 → (∀𝑧𝑣𝑤𝑦 𝑧𝑤 ↔ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))
64, 5anbi12d 608 . . . . . . 7 (𝑣 = 𝐴 → ((𝑦𝑣 ∧ ∀𝑧𝑣𝑤𝑦 𝑧𝑤) ↔ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤)))
76anbi2d 606 . . . . . 6 (𝑣 = 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦𝑣 ∧ ∀𝑧𝑣𝑤𝑦 𝑧𝑤)) ↔ (𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
87exbidv 2001 . . . . 5 (𝑣 = 𝐴 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝑣 ∧ ∀𝑧𝑣𝑤𝑦 𝑧𝑤)) ↔ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))))
98abbidv 2889 . . . 4 (𝑣 = 𝐴 → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝑣 ∧ ∀𝑧𝑣𝑤𝑦 𝑧𝑤))} = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
109inteqd 4614 . . 3 (𝑣 = 𝐴 {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝑣 ∧ ∀𝑧𝑣𝑤𝑦 𝑧𝑤))} = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
11 df-cf 8966 . . 3 cf = (𝑣 ∈ On ↦ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝑣 ∧ ∀𝑧𝑣𝑤𝑦 𝑧𝑤))})
1210, 11fvmptg 6422 . 2 ((𝐴 ∈ On ∧ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))} ∈ V) → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
133, 12mpdan 659 1 (𝐴 ∈ On → (cf‘𝐴) = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦𝐴 ∧ ∀𝑧𝐴𝑤𝑦 𝑧𝑤))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wex 1851  wcel 2144  {cab 2756  wral 3060  wrex 3061  Vcvv 3349  wss 3721   cint 4609  Oncon0 5866  cfv 6031  cardccrd 8960  cfccf 8962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-int 4610  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-cf 8966
This theorem is referenced by:  cfub  9272  cflm  9273  cardcf  9275  cflecard  9276  cfeq0  9279  cfsuc  9280  cff1  9281  cfflb  9282  cfval2  9283  cflim3  9285
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