Theorem fictb 10164

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 8903	. . . . 5 ⊢ (𝐴 ≼ ω → ∃𝑓 𝑓:𝐴–1-1→ω)
2	1	adantl 482	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≼ ω) → ∃𝑓 𝑓:𝐴–1-1→ω)
3		reldom 8896	. . . . . 6 ⊢ Rel ≼
4	3	brrelex2i 5682	. . . . 5 ⊢ (𝐴 ≼ ω → ω ∈ V)
5		omelon2 7826	. . . . . . . . . . 11 ⊢ (ω ∈ V → ω ∈ On)
6	5	ad2antlr 733	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ω ∈ On)
7		pwexg 5314	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ V)
8	7	ad2antrr 732	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝒫 𝐴 ∈ V)
9		inex1g 5254	. . . . . . . . . . . . 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 4073	. . . . . . . . . . . 12 ⊢ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ⊆ (𝒫 𝐴 ∩ Fin)
12		ssdomg 8944	. . . . . . . . . . . 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 6785	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1→ω → 𝑓:𝐴–1-1-onto→ran 𝑓)
15	14	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝑓:𝐴–1-1-onto→ran 𝑓)
16		f1opwfi 9263	. . . . . . . . . . . . . 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 8914	. . . . . . . . . . . . 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 590	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 𝐴 ∩ Fin) ≈ (𝒫 ran 𝑓 ∩ Fin))
20		pwexg 5314	. . . . . . . . . . . . . . . 16 ⊢ (ω ∈ V → 𝒫 ω ∈ V)
21	20	ad2antlr 733	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝒫 ω ∈ V)
22		inex1g 5254	. . . . . . . . . . . . . . 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 6730	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1→ω → 𝑓:𝐴⟶ω)
25	24	frnd 6670	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1→ω → ran 𝑓 ⊆ ω)
26	25	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ran 𝑓 ⊆ ω)
27	26	sspwd 4549	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝒫 ran 𝑓 ⊆ 𝒫 ω)
28	27	ssrind 4179	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ran 𝑓 ∩ Fin) ⊆ (𝒫 ω ∩ Fin))
29		ssdomg 8944	. . . . . . . . . . . . . 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 4572	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑧 → {𝑓} = {𝑧})
32		pweq 4550	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑧 → 𝒫 𝑓 = 𝒫 𝑧)
33	31, 32	xpeq12d 5656	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑧 → ({𝑓} × 𝒫 𝑓) = ({𝑧} × 𝒫 𝑧))
34	33	cbviunv 4975	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓) = ∪ 𝑧 ∈ 𝑥 ({𝑧} × 𝒫 𝑧)
35		iuneq1 4945	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ∪ 𝑧 ∈ 𝑥 ({𝑧} × 𝒫 𝑧) = ∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧))
36	34, 35	eqtrid 2787	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓) = ∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧))
37	36	fveq2d 6838	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓)) = (card‘∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧)))
38	37	cbvmptv 5183	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓))) = (𝑦 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧)))
39	38	ackbij1 10157	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓))):(𝒫 ω ∩ Fin)–1-1-onto→ω
40		f1oeng 8914	. . . . . . . . . . . . . 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 592	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ω ∩ Fin) ≈ ω)
42		domentr 8957	. . . . . . . . . . . . 13 ⊢ (((𝒫 ran 𝑓 ∩ Fin) ≼ (𝒫 ω ∩ Fin) ∧ (𝒫 ω ∩ Fin) ≈ ω) → (𝒫 ran 𝑓 ∩ Fin) ≼ ω)
43	30, 41, 42	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ran 𝑓 ∩ Fin) ≼ ω)
44		endomtr 8956	. . . . . . . . . . . 12 ⊢ (((𝒫 𝐴 ∩ Fin) ≈ (𝒫 ran 𝑓 ∩ Fin) ∧ (𝒫 ran 𝑓 ∩ Fin) ≼ ω) → (𝒫 𝐴 ∩ Fin) ≼ ω)
45	19, 43, 44	syl2anc 590	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 𝐴 ∩ Fin) ≼ ω)
46		domtr 8951	. . . . . . . . . . 11 ⊢ ((((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≼ ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω)
47	13, 45, 46	syl2anc 590	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω)
48		ondomen 9957	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card)
49	6, 47, 48	syl2anc 590	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card)
50		eqid 2740	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦) = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦)
51	50	fifo 9342	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
52	51	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
53		fodomnum 9977	. . . . . . . . 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 8951	. . . . . . . 8 ⊢ (((fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω) → (fi‘𝐴) ≼ ω)
56	54, 47, 55	syl2anc 590	. . . . . . 7 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (fi‘𝐴) ≼ ω)
57	56	ex 413	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ ω ∈ V) → (𝑓:𝐴–1-1→ω → (fi‘𝐴) ≼ ω))
58	57	exlimdv 1940	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ω ∈ V) → (∃𝑓 𝑓:𝐴–1-1→ω → (fi‘𝐴) ≼ ω))
59	4, 58	sylan2 599	. . . 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 413	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ≼ ω → (fi‘𝐴) ≼ ω))
62		fvex 6847	. . . 4 ⊢ (fi‘𝐴) ∈ V
63		ssfii 9329	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ (fi‘𝐴))
64		ssdomg 8944	. . . 4 ⊢ ((fi‘𝐴) ∈ V → (𝐴 ⊆ (fi‘𝐴) → 𝐴 ≼ (fi‘𝐴)))
65	62, 63, 64	mpsyl 68	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≼ (fi‘𝐴))
66		domtr 8951	. . . 4 ⊢ ((𝐴 ≼ (fi‘𝐴) ∧ (fi‘𝐴) ≼ ω) → 𝐴 ≼ ω)
67	66	ex 413	. . 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 213	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ≼ ω ↔ (fi‘𝐴) ≼ ω))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∃wex 1786   ∈ wcel 2119  Vcvv 3432   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  𝒫 cpw 4536  {csn 4562  ∩ cint 4884  ∪ ciun 4928   class class class wbr 5079   ↦ cmpt 5160   × cxp 5623  dom cdm 5625  ran crn 5626   “ cima 5628  Oncon0 6317  –1-1→wf1 6489  –onto→wfo 6490  –1-1-onto→wf1o 6491  ‘cfv 6492  ωcom 7813   ≈ cen 8887   ≼ cdom 8888  Fincfn 8890  ficfi 9320  cardccrd 9857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-oadd 8406  df-er 8640  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fi 9321  df-dju 9823  df-card 9861  df-acn 9864

This theorem is referenced by:  2ndcsb  23439