Theorem cardcf 9663

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 9658	. . . 4 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
2		vex 3444	. . . . . . . . 9 ⊢ 𝑣 ∈ V
3		eqeq1 2802	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → (𝑥 = (card‘𝑦) ↔ 𝑣 = (card‘𝑦)))
4	3	anbi1d 632	. . . . . . . . . 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 1922	. . . . . . . . 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 3615	. . . . . . . 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 6645	. . . . . . . . . . . 12 ⊢ (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘(card‘𝑦)))
8		cardidm 9372	. . . . . . . . . . . 12 ⊢ (card‘(card‘𝑦)) = (card‘𝑦)
9	7, 8	eqtrdi 2849	. . . . . . . . . . 11 ⊢ (𝑣 = (card‘𝑦) → (card‘𝑣) = (card‘𝑦))
10		eqeq2 2810	. . . . . . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝑣 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) → (card‘𝑣) = 𝑣)
13	12	exlimiv 1931	. . . . . . . 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 9357	. . . . . . 7 ⊢ (card‘𝑣) ∈ On
16	14, 15	eqeltrrdi 2899	. . . . . 6 ⊢ (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} → 𝑣 ∈ On)
17	16	ssriv 3919	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ On
18		fvex 6658	. . . . . . 7 ⊢ (cf‘𝐴) ∈ V
19	1, 18	eqeltrrdi 2899	. . . . . 6 ⊢ (𝐴 ∈ On → ∩ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ∈ V)
20		intex 5204	. . . . . 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 7490	. . . . 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 590	. . . 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 2890	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) ∈ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))})
25		fveq2 6645	. . . . 5 ⊢ (𝑣 = (cf‘𝐴) → (card‘𝑣) = (card‘(cf‘𝐴)))
26		id 22	. . . . 5 ⊢ (𝑣 = (cf‘𝐴) → 𝑣 = (cf‘𝐴))
27	25, 26	eqeq12d 2814	. . . 4 ⊢ (𝑣 = (cf‘𝐴) → ((card‘𝑣) = 𝑣 ↔ (card‘(cf‘𝐴)) = (cf‘𝐴)))
28	27, 14	vtoclga 3522	. . 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 17	. 2 ⊢ (𝐴 ∈ On → (card‘(cf‘𝐴)) = (cf‘𝐴))
30		cff 9659	. . . . . 6 ⊢ cf:On⟶On
31	30	fdmi 6498	. . . . 5 ⊢ dom cf = On
32	31	eleq2i 2881	. . . 4 ⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On)
33		ndmfv 6675	. . . 4 ⊢ (¬ 𝐴 ∈ dom cf → (cf‘𝐴) = ∅)
34	32, 33	sylnbir 334	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘𝐴) = ∅)
35		card0 9371	. . . 4 ⊢ (card‘∅) = ∅
36		fveq2 6645	. . . 4 ⊢ ((cf‘𝐴) = ∅ → (card‘(cf‘𝐴)) = (card‘∅))
37		id 22	. . . 4 ⊢ ((cf‘𝐴) = ∅ → (cf‘𝐴) = ∅)
38	35, 36, 37	3eqtr4a 2859	. . 3 ⊢ ((cf‘𝐴) = ∅ → (card‘(cf‘𝐴)) = (cf‘𝐴))
39	34, 38	syl 17	. 2 ⊢ (¬ 𝐴 ∈ On → (card‘(cf‘𝐴)) = (cf‘𝐴))
40	29, 39	pm2.61i 185	1 ⊢ (card‘(cf‘𝐴)) = (cf‘𝐴)

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ∩ cint 4838  dom cdm 5519  Oncon0 6159  ‘cfv 6324  cardccrd 9348  cfccf 9350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-er 8272  df-en 8493  df-card 9352  df-cf 9354

This theorem is referenced by:  cfon  9666  winacard  10103