Theorem fictb 9859

Description: A set is countable iff its collection of finite intersections is countable. (Contributed by Jeff Hankins, 24-Aug-2009.) (Proof shortened by Mario Carneiro, 17-May-2015.)

Assertion

Ref	Expression
fictb	⊢ (𝐴 ∈ 𝐵 → (𝐴 ≼ ω ↔ (fi‘𝐴) ≼ ω))

Proof of Theorem fictb

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

Step	Hyp	Ref	Expression
1		brdomi 8639	. . . . 5 ⊢ (𝐴 ≼ ω → ∃𝑓 𝑓:𝐴–1-1→ω)
2	1	adantl 485	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≼ ω) → ∃𝑓 𝑓:𝐴–1-1→ω)
3		reldom 8632	. . . . . 6 ⊢ Rel ≼
4	3	brrelex2i 5606	. . . . 5 ⊢ (𝐴 ≼ ω → ω ∈ V)
5		omelon2 7657	. . . . . . . . . . 11 ⊢ (ω ∈ V → ω ∈ On)
6	5	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ω ∈ On)
7		pwexg 5271	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ V)
8	7	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝒫 𝐴 ∈ V)
9		inex1g 5212	. . . . . . . . . . . . 13 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
10	8, 9	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 𝐴 ∩ Fin) ∈ V)
11		difss 4046	. . . . . . . . . . . 12 ⊢ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ⊆ (𝒫 𝐴 ∩ Fin)
12		ssdomg 8674	. . . . . . . . . . . 12 ⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → (((𝒫 𝐴 ∩ Fin) ∖ {∅}) ⊆ (𝒫 𝐴 ∩ Fin) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin)))
13	10, 11, 12	mpisyl 21	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin))
14		f1f1orn 6672	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1→ω → 𝑓:𝐴–1-1-onto→ran 𝑓)
15	14	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝑓:𝐴–1-1-onto→ran 𝑓)
16		f1opwfi 8980	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→ran 𝑓 → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝑓 “ 𝑥)):(𝒫 𝐴 ∩ Fin)–1-1-onto→(𝒫 ran 𝑓 ∩ Fin))
17	15, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝑓 “ 𝑥)):(𝒫 𝐴 ∩ Fin)–1-1-onto→(𝒫 ran 𝑓 ∩ Fin))
18		f1oeng 8647	. . . . . . . . . . . . 13 ⊢ (((𝒫 𝐴 ∩ Fin) ∈ V ∧ (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝑓 “ 𝑥)):(𝒫 𝐴 ∩ Fin)–1-1-onto→(𝒫 ran 𝑓 ∩ Fin)) → (𝒫 𝐴 ∩ Fin) ≈ (𝒫 ran 𝑓 ∩ Fin))
19	10, 17, 18	syl2anc 587	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 𝐴 ∩ Fin) ≈ (𝒫 ran 𝑓 ∩ Fin))
20		pwexg 5271	. . . . . . . . . . . . . . . 16 ⊢ (ω ∈ V → 𝒫 ω ∈ V)
21	20	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝒫 ω ∈ V)
22		inex1g 5212	. . . . . . . . . . . . . . 15 ⊢ (𝒫 ω ∈ V → (𝒫 ω ∩ Fin) ∈ V)
23	21, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ω ∩ Fin) ∈ V)
24		f1f 6615	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1→ω → 𝑓:𝐴⟶ω)
25	24	frnd 6553	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1→ω → ran 𝑓 ⊆ ω)
26	25	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ran 𝑓 ⊆ ω)
27	26	sspwd 4528	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝒫 ran 𝑓 ⊆ 𝒫 ω)
28	27	ssrind 4150	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ran 𝑓 ∩ Fin) ⊆ (𝒫 ω ∩ Fin))
29		ssdomg 8674	. . . . . . . . . . . . . 14 ⊢ ((𝒫 ω ∩ Fin) ∈ V → ((𝒫 ran 𝑓 ∩ Fin) ⊆ (𝒫 ω ∩ Fin) → (𝒫 ran 𝑓 ∩ Fin) ≼ (𝒫 ω ∩ Fin)))
30	23, 28, 29	sylc 65	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ran 𝑓 ∩ Fin) ≼ (𝒫 ω ∩ Fin))
31		sneq 4551	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑧 → {𝑓} = {𝑧})
32		pweq 4529	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑧 → 𝒫 𝑓 = 𝒫 𝑧)
33	31, 32	xpeq12d 5582	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑧 → ({𝑓} × 𝒫 𝑓) = ({𝑧} × 𝒫 𝑧))
34	33	cbviunv 4949	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓) = ∪ 𝑧 ∈ 𝑥 ({𝑧} × 𝒫 𝑧)
35		iuneq1 4920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ∪ 𝑧 ∈ 𝑥 ({𝑧} × 𝒫 𝑧) = ∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧))
36	34, 35	eqtrid 2789	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓) = ∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧))
37	36	fveq2d 6721	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓)) = (card‘∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧)))
38	37	cbvmptv 5158	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓))) = (𝑦 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧)))
39	38	ackbij1 9852	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓))):(𝒫 ω ∩ Fin)–1-1-onto→ω
40		f1oeng 8647	. . . . . . . . . . . . . 14 ⊢ (((𝒫 ω ∩ Fin) ∈ V ∧ (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓))):(𝒫 ω ∩ Fin)–1-1-onto→ω) → (𝒫 ω ∩ Fin) ≈ ω)
41	23, 39, 40	sylancl 589	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ω ∩ Fin) ≈ ω)
42		domentr 8687	. . . . . . . . . . . . 13 ⊢ (((𝒫 ran 𝑓 ∩ Fin) ≼ (𝒫 ω ∩ Fin) ∧ (𝒫 ω ∩ Fin) ≈ ω) → (𝒫 ran 𝑓 ∩ Fin) ≼ ω)
43	30, 41, 42	syl2anc 587	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ran 𝑓 ∩ Fin) ≼ ω)
44		endomtr 8686	. . . . . . . . . . . 12 ⊢ (((𝒫 𝐴 ∩ Fin) ≈ (𝒫 ran 𝑓 ∩ Fin) ∧ (𝒫 ran 𝑓 ∩ Fin) ≼ ω) → (𝒫 𝐴 ∩ Fin) ≼ ω)
45	19, 43, 44	syl2anc 587	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 𝐴 ∩ Fin) ≼ ω)
46		domtr 8681	. . . . . . . . . . 11 ⊢ ((((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≼ ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω)
47	13, 45, 46	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω)
48		ondomen 9651	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card)
49	6, 47, 48	syl2anc 587	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card)
50		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦) = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦)
51	50	fifo 9048	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
52	51	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
53		fodomnum 9671	. . . . . . . . 9 ⊢ (((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card → ((𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴) → (fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅})))
54	49, 52, 53	sylc 65	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
55		domtr 8681	. . . . . . . 8 ⊢ (((fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω) → (fi‘𝐴) ≼ ω)
56	54, 47, 55	syl2anc 587	. . . . . . 7 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (fi‘𝐴) ≼ ω)
57	56	ex 416	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ ω ∈ V) → (𝑓:𝐴–1-1→ω → (fi‘𝐴) ≼ ω))
58	57	exlimdv 1941	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ω ∈ V) → (∃𝑓 𝑓:𝐴–1-1→ω → (fi‘𝐴) ≼ ω))
59	4, 58	sylan2 596	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≼ ω) → (∃𝑓 𝑓:𝐴–1-1→ω → (fi‘𝐴) ≼ ω))
60	2, 59	mpd 15	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≼ ω) → (fi‘𝐴) ≼ ω)
61	60	ex 416	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ≼ ω → (fi‘𝐴) ≼ ω))
62		fvex 6730	. . . 4 ⊢ (fi‘𝐴) ∈ V
63		ssfii 9035	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ (fi‘𝐴))
64		ssdomg 8674	. . . 4 ⊢ ((fi‘𝐴) ∈ V → (𝐴 ⊆ (fi‘𝐴) → 𝐴 ≼ (fi‘𝐴)))
65	62, 63, 64	mpsyl 68	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≼ (fi‘𝐴))
66		domtr 8681	. . . 4 ⊢ ((𝐴 ≼ (fi‘𝐴) ∧ (fi‘𝐴) ≼ ω) → 𝐴 ≼ ω)
67	66	ex 416	. . 3 ⊢ (𝐴 ≼ (fi‘𝐴) → ((fi‘𝐴) ≼ ω → 𝐴 ≼ ω))
68	65, 67	syl 17	. 2 ⊢ (𝐴 ∈ 𝐵 → ((fi‘𝐴) ≼ ω → 𝐴 ≼ ω))
69	61, 68	impbid 215	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ≼ ω ↔ (fi‘𝐴) ≼ ω))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∃wex 1787   ∈ wcel 2110  Vcvv 3408   ∖ cdif 3863   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  𝒫 cpw 4513  {csn 4541  ∩ cint 4859  ∪ ciun 4904   class class class wbr 5053   ↦ cmpt 5135   × cxp 5549  dom cdm 5551  ran crn 5552   “ cima 5554  Oncon0 6213  –1-1→wf1 6377  –onto→wfo 6378  –1-1-onto→wf1o 6379  ‘cfv 6380  ωcom 7644   ≈ cen 8623   ≼ cdom 8624  Fincfn 8626  ficfi 9026  cardccrd 9551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-oadd 8206  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fi 9027  df-dju 9517  df-card 9555  df-acn 9558

This theorem is referenced by:  2ndcsb  22346