Theorem fictb 10181

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 8898	. . . . 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 8889	. . . . . 6 ⊢ Rel ≼
4	3	brrelex2i 5689	. . . . 5 ⊢ (𝐴 ≼ ω → ω ∈ V)
5		omelon2 7815	. . . . . . . . . . 11 ⊢ (ω ∈ V → ω ∈ On)
6	5	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ω ∈ On)
7		pwexg 5333	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ V)
8	7	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝒫 𝐴 ∈ V)
9		inex1g 5276	. . . . . . . . . . . . 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 4091	. . . . . . . . . . . 12 ⊢ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ⊆ (𝒫 𝐴 ∩ Fin)
12		ssdomg 8940	. . . . . . . . . . . 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 6795	. . . . . . . . . . . . . . 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 9300	. . . . . . . . . . . . . 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 8911	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 𝐴 ∩ Fin) ≈ (𝒫 ran 𝑓 ∩ Fin))
20		pwexg 5333	. . . . . . . . . . . . . . . 16 ⊢ (ω ∈ V → 𝒫 ω ∈ V)
21	20	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝒫 ω ∈ V)
22		inex1g 5276	. . . . . . . . . . . . . . 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 6738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1→ω → 𝑓:𝐴⟶ω)
25	24	frnd 6676	. . . . . . . . . . . . . . . . 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 4573	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝒫 ran 𝑓 ⊆ 𝒫 ω)
28	27	ssrind 4195	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ran 𝑓 ∩ Fin) ⊆ (𝒫 ω ∩ Fin))
29		ssdomg 8940	. . . . . . . . . . . . . 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 4596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑧 → {𝑓} = {𝑧})
32		pweq 4574	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑧 → 𝒫 𝑓 = 𝒫 𝑧)
33	31, 32	xpeq12d 5664	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑧 → ({𝑓} × 𝒫 𝑓) = ({𝑧} × 𝒫 𝑧))
34	33	cbviunv 5000	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓) = ∪ 𝑧 ∈ 𝑥 ({𝑧} × 𝒫 𝑧)
35		iuneq1 4970	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ∪ 𝑧 ∈ 𝑥 ({𝑧} × 𝒫 𝑧) = ∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧))
36	34, 35	eqtrid 2788	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓) = ∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧))
37	36	fveq2d 6846	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓)) = (card‘∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧)))
38	37	cbvmptv 5218	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓))) = (𝑦 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧)))
39	38	ackbij1 10174	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓))):(𝒫 ω ∩ Fin)–1-1-onto→ω
40		f1oeng 8911	. . . . . . . . . . . . . 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 586	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ω ∩ Fin) ≈ ω)
42		domentr 8953	. . . . . . . . . . . . 13 ⊢ (((𝒫 ran 𝑓 ∩ Fin) ≼ (𝒫 ω ∩ Fin) ∧ (𝒫 ω ∩ Fin) ≈ ω) → (𝒫 ran 𝑓 ∩ Fin) ≼ ω)
43	30, 41, 42	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ran 𝑓 ∩ Fin) ≼ ω)
44		endomtr 8952	. . . . . . . . . . . 12 ⊢ (((𝒫 𝐴 ∩ Fin) ≈ (𝒫 ran 𝑓 ∩ Fin) ∧ (𝒫 ran 𝑓 ∩ Fin) ≼ ω) → (𝒫 𝐴 ∩ Fin) ≼ ω)
45	19, 43, 44	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 𝐴 ∩ Fin) ≼ ω)
46		domtr 8947	. . . . . . . . . . 11 ⊢ ((((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≼ ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω)
47	13, 45, 46	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω)
48		ondomen 9973	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card)
49	6, 47, 48	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card)
50		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦) = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦)
51	50	fifo 9368	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
52	51	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
53		fodomnum 9993	. . . . . . . . 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 8947	. . . . . . . 8 ⊢ (((fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω) → (fi‘𝐴) ≼ ω)
56	54, 47, 55	syl2anc 584	. . . . . . 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 1936	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ω ∈ V) → (∃𝑓 𝑓:𝐴–1-1→ω → (fi‘𝐴) ≼ ω))
59	4, 58	sylan2 593	. . . 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 6855	. . . 4 ⊢ (fi‘𝐴) ∈ V
63		ssfii 9355	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ (fi‘𝐴))
64		ssdomg 8940	. . . 4 ⊢ ((fi‘𝐴) ∈ V → (𝐴 ⊆ (fi‘𝐴) → 𝐴 ≼ (fi‘𝐴)))
65	62, 63, 64	mpsyl 68	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≼ (fi‘𝐴))
66		domtr 8947	. . . 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 211	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ≼ ω ↔ (fi‘𝐴) ≼ ω))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∃wex 1781   ∈ wcel 2106  Vcvv 3445   ∖ cdif 3907   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  𝒫 cpw 4560  {csn 4586  ∩ cint 4907  ∪ ciun 4954   class class class wbr 5105   ↦ cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634   “ cima 5636  Oncon0 6317  –1-1→wf1 6493  –onto→wfo 6494  –1-1-onto→wf1o 6495  ‘cfv 6496  ωcom 7802   ≈ cen 8880   ≼ cdom 8881  Fincfn 8883  ficfi 9346  cardccrd 9871

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9347  df-dju 9837  df-card 9875  df-acn 9878

This theorem is referenced by:  2ndcsb  22800