Theorem cflim3 10241

Description: Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)

Hypothesis

Ref	Expression
cflim3.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
cflim3	⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem cflim3

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

Step	Hyp	Ref	Expression
1		limord 6422	. . . 4 ⊢ (Lim 𝐴 → Ord 𝐴)
2		cflim3.1	. . . . 5 ⊢ 𝐴 ∈ V
3	2	elon 6369	. . . 4 ⊢ (𝐴 ∈ On ↔ Ord 𝐴)
4	1, 3	sylibr 237	. . 3 ⊢ (Lim 𝐴 → 𝐴 ∈ On)
5		cfval 10225	. . 3 ⊢ (𝐴 ∈ On → (cf‘𝐴) = ∩ {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))})
6	4, 5	syl 18	. 2 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))})
7		fvex 6894	. . . 4 ⊢ (card‘𝑥) ∈ V
8	7	dfiin2 4997	. . 3 ⊢ ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥) = ∩ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)}
9		df-rex 3090	. . . . . 6 ⊢ (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)))
10		ancom 465	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ (𝑦 = (card‘𝑥) ∧ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}))
11		rabid 3437	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴))
12		velpw 4567	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
13	12	anbi1i 635	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ ∪ 𝑥 = 𝐴))
14		coflim 10240	. . . . . . . . . . . 12 ⊢ ((Lim 𝐴 ∧ 𝑥 ⊆ 𝐴) → (∪ 𝑥 = 𝐴 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))
15	14	pm5.32da 589	. . . . . . . . . . 11 ⊢ (Lim 𝐴 → ((𝑥 ⊆ 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤)))
16	13, 15	bitrid 286	. . . . . . . . . 10 ⊢ (Lim 𝐴 → ((𝑥 ∈ 𝒫 𝐴 ∧ ∪ 𝑥 = 𝐴) ↔ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤)))
17	11, 16	bitrid 286	. . . . . . . . 9 ⊢ (Lim 𝐴 → (𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ↔ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤)))
18	17	anbi2d 641	. . . . . . . 8 ⊢ (Lim 𝐴 → ((𝑦 = (card‘𝑥) ∧ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}) ↔ (𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))))
19	10, 18	bitrid 286	. . . . . . 7 ⊢ (Lim 𝐴 → ((𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ (𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))))
20	19	exbidv 1951	. . . . . 6 ⊢ (Lim 𝐴 → (∃𝑥(𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} ∧ 𝑦 = (card‘𝑥)) ↔ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))))
21	9, 20	bitrid 286	. . . . 5 ⊢ (Lim 𝐴 → (∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥) ↔ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))))
22	21	abbidv 2829	. . . 4 ⊢ (Lim 𝐴 → {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} = {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))})
23	22	inteqd 4917	. . 3 ⊢ (Lim 𝐴 → ∩ {𝑦 ∣ ∃𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴}𝑦 = (card‘𝑥)} = ∩ {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))})
24	8, 23	eqtr2id 2811	. 2 ⊢ (Lim 𝐴 → ∩ {𝑦 ∣ ∃𝑥(𝑦 = (card‘𝑥) ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑥 𝑧 ⊆ 𝑤))} = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))
25	6, 24	eqtrd 2798	1 ⊢ (Lim 𝐴 → (cf‘𝐴) = ∩ 𝑥 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∪ 𝑥 = 𝐴} (card‘𝑥))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4562  ∪ cuni 4872  ∩ cint 4912  ∩ ciin 4957  Ord word 6359  Oncon0 6360  Lim wlim 6361  ‘cfv 6536  cardccrd 9917  cfccf 9919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-ord 6363  df-on 6364  df-lim 6365  df-iota 6492  df-fun 6538  df-fv 6544  df-cf 9923

This theorem is referenced by:  cflim2  10242  cfss  10244  cfslb  10245