Theorem fictb 10154

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 8896	. . . . 5 ⊢ (𝐴 ≼ ω → ∃𝑓 𝑓:𝐴–1-1→ω)
2	1	adantl 481	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≼ ω) → ∃𝑓 𝑓:𝐴–1-1→ω)
3		reldom 8889	. . . . . 6 ⊢ Rel ≼
4	3	brrelex2i 5681	. . . . 5 ⊢ (𝐴 ≼ ω → ω ∈ V)
5		omelon2 7821	. . . . . . . . . . 11 ⊢ (ω ∈ V → ω ∈ On)
6	5	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ω ∈ On)
7		pwexg 5323	. . . . . . . . . . . . . 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 5264	. . . . . . . . . . . . 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 4088	. . . . . . . . . . . 12 ⊢ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ⊆ (𝒫 𝐴 ∩ Fin)
12		ssdomg 8937	. . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝑓:𝐴–1-1-onto→ran 𝑓)
16		f1opwfi 9256	. . . . . . . . . . . . . 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 8907	. . . . . . . . . . . . 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 5323	. . . . . . . . . . . . . . . 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 5264	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ran 𝑓 ⊆ ω)
27	26	sspwd 4567	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝒫 ran 𝑓 ⊆ 𝒫 ω)
28	27	ssrind 4196	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ran 𝑓 ∩ Fin) ⊆ (𝒫 ω ∩ Fin))
29		ssdomg 8937	. . . . . . . . . . . . . 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 4590	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑧 → {𝑓} = {𝑧})
32		pweq 4568	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑧 → 𝒫 𝑓 = 𝒫 𝑧)
33	31, 32	xpeq12d 5655	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑧 → ({𝑓} × 𝒫 𝑓) = ({𝑧} × 𝒫 𝑧))
34	33	cbviunv 4994	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓) = ∪ 𝑧 ∈ 𝑥 ({𝑧} × 𝒫 𝑧)
35		iuneq1 4963	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ∪ 𝑧 ∈ 𝑥 ({𝑧} × 𝒫 𝑧) = ∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧))
36	34, 35	eqtrid 2783	. . . . . . . . . . . . . . . . 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 5202	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓))) = (𝑦 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧)))
39	38	ackbij1 10147	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓))):(𝒫 ω ∩ Fin)–1-1-onto→ω
40		f1oeng 8907	. . . . . . . . . . . . . 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 8950	. . . . . . . . . . . . 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 8949	. . . . . . . . . . . 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 8944	. . . . . . . . . . 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 9947	. . . . . . . . . 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 9335	. . . . . . . . . 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 9967	. . . . . . . . 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 8944	. . . . . . . 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 412	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ ω ∈ V) → (𝑓:𝐴–1-1→ω → (fi‘𝐴) ≼ ω))
58	57	exlimdv 1934	. . . . 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 412	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ≼ ω → (fi‘𝐴) ≼ ω))
62		fvex 6847	. . . 4 ⊢ (fi‘𝐴) ∈ V
63		ssfii 9322	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ (fi‘𝐴))
64		ssdomg 8937	. . . 4 ⊢ ((fi‘𝐴) ∈ V → (𝐴 ⊆ (fi‘𝐴) → 𝐴 ≼ (fi‘𝐴)))
65	62, 63, 64	mpsyl 68	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≼ (fi‘𝐴))
66		domtr 8944	. . . 4 ⊢ ((𝐴 ≼ (fi‘𝐴) ∧ (fi‘𝐴) ≼ ω) → 𝐴 ≼ ω)
67	66	ex 412	. . 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 212	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ≼ ω ↔ (fi‘𝐴) ≼ ω))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1780   ∈ wcel 2113  Vcvv 3440   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580  ∩ cint 4902  ∪ ciun 4946   class class class wbr 5098   ↦ cmpt 5179   × cxp 5622  dom cdm 5624  ran crn 5625   “ cima 5627  Oncon0 6317  –1-1→wf1 6489  –onto→wfo 6490  –1-1-onto→wf1o 6491  ‘cfv 6492  ωcom 7808   ≈ cen 8880   ≼ cdom 8881  Fincfn 8883  ficfi 9313  cardccrd 9847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fi 9314  df-dju 9813  df-card 9851  df-acn 9854

This theorem is referenced by:  2ndcsb  23393