Theorem cfval 10148

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 10146	. . 3 ⊢ (𝐴 ∈ On → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
2		intexab 5288	. . 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 218	. 2 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ∈ V)
4		sseq2 3958	. . . . . . . 8 ⊢ (𝑣 = 𝐴 → (𝑦 ⊆ 𝑣 ↔ 𝑦 ⊆ 𝐴))
5		raleq 3291	. . . . . . . 8 ⊢ (𝑣 = 𝐴 → (∀𝑧 ∈ 𝑣 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))
6	4, 5	anbi12d 632	. . . . . . 7 ⊢ (𝑣 = 𝐴 → ((𝑦 ⊆ 𝑣 ∧ ∀𝑧 ∈ 𝑣 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
7	6	anbi2d 630	. . . . . 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 1922	. . . . 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 2799	. . . 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 4904	. . 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 9844	. . 3 ⊢ cf = (𝑣 ∈ On ↦ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝑣 ∧ ∀𝑧 ∈ 𝑣 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
12	10, 11	fvmptg 6936	. 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 687	1 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2711  ∀wral 3049  ∃wrex 3058  Vcvv 3438   ⊆ wss 3899  ∩ cint 4899  Oncon0 6314  ‘cfv 6489  cardccrd 9838  cfccf 9840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-int 4900  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-cf 9844

This theorem is referenced by:  cfub  10150  cflm  10151  cardcf  10153  cflecard  10154  cfeq0  10157  cfsuc  10158  cff1  10159  cfflb  10160  cfval2  10161  cflim3  10163