Theorem cardcf 10329

Description: Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)

Assertion

Ref	Expression
cardcf	⊢ (card‘(cf‘𝐴)) = (cf‘𝐴)

Proof of Theorem cardcf

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

Step	Hyp	Ref	Expression
1		cfval 10324	. . . 4 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
2		vex 3454	. . . . . . . . 9 ⊢ 𝑣 ∈ V
3		eqeq1 2764	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → (𝑥 = (card‘𝑦) ↔ 𝑣 = (card‘𝑦)))
4	3	anbi1d 643	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (𝑣 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
5	4	exbidv 1953	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → (∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))))
6	2, 5	elab 3632	. . . . . . . 8 ⊢ (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ↔ ∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))
7		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘(card‘𝑦)))
8		cardidm 10040	. . . . . . . . . . . 12 ⊢ (card‘(card‘𝑦)) = (card‘𝑦)
9	7, 8	eqtrdi 2811	. . . . . . . . . . 11 ⊢ (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘𝑦))
10		eqeq2 2772	. . . . . . . . . . 11 ⊢ (𝑣 = (card‘𝑦) → ((card‘𝑣) = 𝑣 ↔ (card‘𝑣) = (card‘𝑦)))
11	9, 10	mpbird 260	. . . . . . . . . 10 ⊢ (𝑣 = (card‘𝑦) → (card‘𝑣) = 𝑣)
12	11	adantr 486	. . . . . . . . 9 ⊢ ((𝑣 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (card‘𝑣) = 𝑣)
13	12	exlimiv 1962	. . . . . . . 8 ⊢ (∃𝑦(𝑣 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (card‘𝑣) = 𝑣)
14	6, 13	sylbi 220	. . . . . . 7 ⊢ (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} → (card‘𝑣) = 𝑣)
15		cardon 10025	. . . . . . 7 ⊢ (card‘𝑣) ∈ On
16	14, 15	eqeltrrdi 2869	. . . . . 6 ⊢ (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} → 𝑣 ∈ On)
17	16	ssriv 3912	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ On
18		fvex 6898	. . . . . . 7 ⊢ (cf‘𝐴) ∈ V
19	1, 18	eqeltrrdi 2869	. . . . . 6 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ∈ V)
20		intex 5304	. . . . . 6 ⊢ ({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)
21	19, 20	sylibr 237	. . . . 5 ⊢ (𝐴 ∈ On → {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ≠ ∅)
22		onint 7804	. . . . 5 ⊢ (({𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ On ∧ {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))} ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
23	17, 21, 22	sylancr 599	. . . 4 ⊢ (𝐴 ∈ On → ∩ {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))})
24	1, 23	eqeltrd 2860	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
25		fveq2 6885	. . . . 5 ⊢ (𝑣 = (cf‘𝐴) → (card‘𝑣) = (card‘(cf‘𝐴)))
26		id 23	. . . . 5 ⊢ (𝑣 = (cf‘𝐴) → 𝑣 = (cf‘𝐴))
27	25, 26	eqeq12d 2776	. . . 4 ⊢ (𝑣 = (cf‘𝐴) → ((card‘𝑣) = 𝑣 ↔ (card‘(cf‘𝐴)) = (cf‘𝐴)))
28	27, 14	vtoclga 3536	. . 3 ⊢ ((cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} → (card‘(cf‘𝐴)) = (cf‘𝐴))
29	24, 28	syl 18	. 2 ⊢ (𝐴 ∈ On → (card‘(cf‘𝐴)) = (cf‘𝐴))
30		cff 10325	. . . . . 6 ⊢ cf:On⟶On
31	30	fdmi 6721	. . . . 5 ⊢ dom cf = On
32	31	eleq2i 2852	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
33		ndmfv 6917	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
34	32, 33	sylnbir 334	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
35		card0 10039	. . . 4 ⊢ (card‘∅) = ∅
36		fveq2 6885	. . . 4 ⊢ ((cf‘𝐴) = ∅ → (card‘(cf‘𝐴)) = (card‘∅))
37		id 23	. . . 4 ⊢ ((cf‘𝐴) = ∅ → (cf‘𝐴) = ∅)
38	35, 36, 37	3eqtr4a 2821	. . 3 ⊢ ((cf‘𝐴) = ∅ → (card‘(cf‘𝐴)) = (cf‘𝐴))
39	34, 38	syl 18	. 2 ⊢ (¬ 𝐴 ∈ On → (card‘(cf‘𝐴)) = (cf‘𝐴))
40	29, 39	pm2.61i 184	1 ⊢ (card‘(cf‘𝐴)) = (cf‘𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  ∩ cint 4906  dom cdm 5650  Oncon0 6362  ‘cfv 6538  cardccrd 10016  cfccf 10018

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-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  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-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-er 8717  df-en 8974  df-card 10020  df-cf 10022

This theorem is used by:  cfonOLD  10333  winacard  10777