Theorem cflm 10366

Description: Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257. (Contributed by NM, 26-Apr-2004.)

Assertion

Ref	Expression
cflm	⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))})

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝐵(𝑥, 𝑦)

Proof of Theorem cflm

Dummy variables 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3475	. 2 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V)
2		limsuc 7894	. . . . . . . . . . . . . . . . . 18 ⊢ (Lim 𝐴 → (𝑣 ∈ 𝐴 ↔ suc 𝑣 ∈ 𝐴))
3	2	biimpd 232	. . . . . . . . . . . . . . . . 17 ⊢ (Lim 𝐴 → (𝑣 ∈ 𝐴 → suc 𝑣 ∈ 𝐴))
4		sseq1 3938	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = suc 𝑣 → (𝑧 ⊆ 𝑤 ↔ suc 𝑣 ⊆ 𝑤))
5	4	rexbidv 3190	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = suc 𝑣 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝑦 suc 𝑣 ⊆ 𝑤))
6	5	rspcv 3576	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝑣 ∈ 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → ∃𝑤 ∈ 𝑦 suc 𝑣 ⊆ 𝑤))
7		sucssel 6486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ V → (suc 𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑤))
8	7	elv 3459	. . . . . . . . . . . . . . . . . . . 20 ⊢ (suc 𝑣 ⊆ 𝑤 → 𝑣 ∈ 𝑤)
9	8	reximi 3104	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑤 ∈ 𝑦 suc 𝑣 ⊆ 𝑤 → ∃𝑤 ∈ 𝑦 𝑣 ∈ 𝑤)
10		eluni2 4876	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ ∪ 𝑦 ↔ ∃𝑤 ∈ 𝑦 𝑣 ∈ 𝑤)
11	9, 10	sylibr 237	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑤 ∈ 𝑦 suc 𝑣 ⊆ 𝑤 → 𝑣 ∈ ∪ 𝑦)
12	6, 11	syl6com 38	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → (suc 𝑣 ∈ 𝐴 → 𝑣 ∈ ∪ 𝑦))
13	3, 12	syl9 78	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → (𝑣 ∈ 𝐴 → 𝑣 ∈ ∪ 𝑦)))
14	13	ralrimdv 3164	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → ∀𝑣 ∈ 𝐴 𝑣 ∈ ∪ 𝑦))
15		dfss3 3902	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ ∪ 𝑦 ↔ ∀𝑣 ∈ 𝐴 𝑣 ∈ ∪ 𝑦)
16	14, 15	imbitrrdi 255	. . . . . . . . . . . . . 14 ⊢ (Lim 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → 𝐴 ⊆ ∪ 𝑦))
17	16	adantr 486	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → 𝐴 ⊆ ∪ 𝑦))
18		uniss 4880	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ 𝐴 → ∪ 𝑦 ⊆ ∪ 𝐴)
19		limuni 6451	. . . . . . . . . . . . . . . 16 ⊢ (Lim 𝐴 → 𝐴 = ∪ 𝐴)
20	19	sseq2d 3945	. . . . . . . . . . . . . . 15 ⊢ (Lim 𝐴 → (∪ 𝑦 ⊆ 𝐴 ↔ ∪ 𝑦 ⊆ ∪ 𝐴))
21	18, 20	imbitrrid 249	. . . . . . . . . . . . . 14 ⊢ (Lim 𝐴 → (𝑦 ⊆ 𝐴 → ∪ 𝑦 ⊆ 𝐴))
22	21	imp 412	. . . . . . . . . . . . 13 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴) → ∪ 𝑦 ⊆ 𝐴)
23	17, 22	jctird 536	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → (𝐴 ⊆ ∪ 𝑦 ∧ ∪ 𝑦 ⊆ 𝐴)))
24		eqss 3928	. . . . . . . . . . . 12 ⊢ (𝐴 = ∪ 𝑦 ↔ (𝐴 ⊆ ∪ 𝑦 ∧ ∪ 𝑦 ⊆ 𝐴))
25	23, 24	imbitrrdi 255	. . . . . . . . . . 11 ⊢ ((Lim 𝐴 ∧ 𝑦 ⊆ 𝐴) → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 → 𝐴 = ∪ 𝑦))
26	25	imdistanda 582	. . . . . . . . . 10 ⊢ (Lim 𝐴 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) → (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦)))
27	26	anim2d 624	. . . . . . . . 9 ⊢ (Lim 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))))
28	27	eximdv 1952	. . . . . . . 8 ⊢ (Lim 𝐴 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))))
29	28	ss2abdv 3993	. . . . . . 7 ⊢ (Lim 𝐴 → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))})
30		intss 4934	. . . . . . 7 ⊢ ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
31	29, 30	syl 18	. . . . . 6 ⊢ (Lim 𝐴 → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
32	31	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
33		limelon 6454	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
34		cfval 10363	. . . . . 6 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
35	33, 34	syl 18	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
36	32, 35	sseqtrrd 3950	. . . 4 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ (cf‘𝐴))
37		cfub 10365	. . . . 5 ⊢ (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}
38		eqimss 3971	. . . . . . . . . 10 ⊢ (𝐴 = ∪ 𝑦 → 𝐴 ⊆ ∪ 𝑦)
39	38	anim2i 629	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦) → (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))
40	39	anim2i 629	. . . . . . . 8 ⊢ ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦)) → (𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦)))
41	40	eximi 1868	. . . . . . 7 ⊢ (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦)))
42	41	ss2abi 3994	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))}
43		intss 4934	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))} → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))} ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))})
44	42, 43	ax-mp 5	. . . . 5 ⊢ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))} ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))}
45	37, 44	sstri 3922	. . . 4 ⊢ (cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))}
46	36, 45	jctil 529	. . 3 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ∧ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ (cf‘𝐴)))
47		eqss 3928	. . 3 ⊢ ((cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ↔ ((cf‘𝐴) ⊆ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ∧ ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))} ⊆ (cf‘𝐴)))
48	46, 47	sylibr 237	. 2 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))})
49	1, 48	sylan 592	1 ⊢ ((𝐴 ∈ 𝐵 ∧ Lim 𝐴) → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ 𝐴 = ∪ 𝑦))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ∪ cuni 4872  ∩ cint 4912  Oncon0 6388  Lim wlim 6389  suc csuc 6390  ‘cfv 6566  cardccrd 10053  cfccf 10055

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-card 10057  df-cf 10059

This theorem is used by:  gruina  10942