Theorem cfval 10327

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) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})

Distinct variable group:   𝑤,𝐴,𝑥,𝑦,𝑧

Proof of Theorem cfval

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cflem 10326	. . 3 ⊢ (𝐴 ∈ On → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
2		intexab 5306	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ∈ V)
3	1, 2	sylib 221	. 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ∈ V)
4		sseq2 3934	. . . . . . . 8 ⊢ (𝑣 = 𝐴 → (𝑦 ⊆ 𝑣 ↔ 𝑦 ⊆ 𝐴))
5		raleq 3316	. . . . . . . 8 ⊢ (𝑣 = 𝐴 → (∀𝑧 ∈ 𝑣 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
6	4, 5	anbi12d 644	. . . . . . 7 ⊢ (𝑣 = 𝐴 → ((𝑦 ⊆ 𝑣 ∧ ∀𝑧 ∈ 𝑣 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
7	6	anbi2d 642	. . . . . 6 ⊢ (𝑣 = 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝑣 ∧ ∀𝑧 ∈ 𝑣 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
8	7	exbidv 1953	. . . . 5 ⊢ (𝑣 = 𝐴 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝑣 ∧ ∀𝑧 ∈ 𝑣 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
9	8	abbidv 2826	. . . 4 ⊢ (𝑣 = 𝐴 → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝑣 ∧ ∀𝑧 ∈ 𝑣 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} = {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
10	9	inteqd 4911	. . 3 ⊢ (𝑣 = 𝐴 → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝑣 ∧ ∀𝑧 ∈ 𝑣 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
11		df-cf 10025	. . 3 ⊢ cf = (𝑣 ∈ On ↦ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝑣 ∧ ∀𝑧 ∈ 𝑣 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
12	10, 11	fvmptg 6991	. 2 ⊢ ((𝐴 ∈ On ∧ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ∈ V) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
13	3, 12	mpdan 700	1 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  ∩ cint 4906  Oncon0 6362  ‘cfv 6538  cardccrd 10019  cfccf 10021

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-cf 10025

This theorem is used by:  cfub  10329  cflm  10330  cardcf  10332  cflecard  10333  cfeq0  10337  cfsuc  10338  cff1  10339  cfflb  10340  cfval2  10341  cflim3  10343