Theorem cfval 10144

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 10142	. . 3 ⊢ (𝐴 ∈ On → ∃𝑥∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
2		intexab 5286	. . 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 3956	. . . . . . . 8 ⊢ (𝑣 = 𝐴 → (𝑦 ⊆ 𝑣 ↔ 𝑦 ⊆ 𝐴))
5		raleq 3289	. . . . . . . 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 2797	. . . 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 4902	. . 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 9840	. . 3 ⊢ cf = (𝑣 ∈ On ↦ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝑣 ∧ ∀𝑧 ∈ 𝑣 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
12	10, 11	fvmptg 6933	. 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 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ⊆ wss 3897  ∩ cint 4897  Oncon0 6312  ‘cfv 6487  cardccrd 9834  cfccf 9836

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6443  df-fun 6489  df-fv 6495  df-cf 9840

This theorem is referenced by:  cfub  10146  cflm  10147  cardcf  10149  cflecard  10150  cfeq0  10153  cfsuc  10154  cff1  10155  cfflb  10156  cfval2  10157  cflim3  10159