Theorem fictb 10303

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 8970	. . . . 5 ⊢ (𝐴 ≼ ω → ∃𝑓 𝑓:𝐴–1-1→ω)
2	1	adantl 487	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≼ ω) → ∃𝑓 𝑓:𝐴–1-1→ω)
3		reldom 8963	. . . . . 6 ⊢ Rel ≼
4	3	brrelex2i 5708	. . . . 5 ⊢ (𝐴 ≼ ω → ω ∈ V)
5		omelon2 7879	. . . . . . . . . . 11 ⊢ (ω ∈ V → ω ∈ On)
6	5	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ω ∈ On)
7		pwexg 5340	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ 𝐵 → 𝒫 𝐴 ∈ V)
8	7	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝒫 𝐴 ∈ V)
9		inex1g 5279	. . . . . . . . . . . . 13 ⊢ (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∩ Fin) ∈ V)
10	8, 9	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 𝐴 ∩ Fin) ∈ V)
11		difss 4083	. . . . . . . . . . . 12 ⊢ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ⊆ (𝒫 𝐴 ∩ Fin)
12		ssdomg 9011	. . . . . . . . . . . 12 ⊢ ((𝒫 𝐴 ∩ Fin) ∈ V → (((𝒫 𝐴 ∩ Fin) ∖ {∅}) ⊆ (𝒫 𝐴 ∩ Fin) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin)))
13	10, 11, 12	mpisyl 22	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin))
14		f1f1orn 6828	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝐴–1-1→ω → 𝑓:𝐴–1-1-onto→ran 𝑓)
15	14	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝑓:𝐴–1-1-onto→ran 𝑓)
16		f1opwfi 9329	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . 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 8981	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 𝐴 ∩ Fin) ≈ (𝒫 ran 𝑓 ∩ Fin))
20		pwexg 5340	. . . . . . . . . . . . . . . 16 ⊢ (ω ∈ V → 𝒫 ω ∈ V)
21	20	ad2antlr 740	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝒫 ω ∈ V)
22		inex1g 5279	. . . . . . . . . . . . . . 15 ⊢ (𝒫 ω ∈ V → (𝒫 ω ∩ Fin) ∈ V)
23	21, 22	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ω ∩ Fin) ∈ V)
24		f1f 6770	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:𝐴–1-1→ω → 𝑓:𝐴⟶ω)
25	24	frnd 6710	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:𝐴–1-1→ω → ran 𝑓 ⊆ ω)
26	25	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ran 𝑓 ⊆ ω)
27	26	sspwd 4570	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → 𝒫 ran 𝑓 ⊆ 𝒫 ω)
28	27	ssrind 4189	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ran 𝑓 ∩ Fin) ⊆ (𝒫 ω ∩ Fin))
29		ssdomg 9011	. . . . . . . . . . . . . 14 ⊢ ((𝒫 ω ∩ Fin) ∈ V → ((𝒫 ran 𝑓 ∩ Fin) ⊆ (𝒫 ω ∩ Fin) → (𝒫 ran 𝑓 ∩ Fin) ≼ (𝒫 ω ∩ Fin)))
30	23, 28, 29	sylc 66	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ran 𝑓 ∩ Fin) ≼ (𝒫 ω ∩ Fin))
31		sneq 4594	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑧 → {𝑓} = {𝑧})
32		pweq 4571	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑧 → 𝒫 𝑓 = 𝒫 𝑧)
33	31, 32	xpeq12d 5682	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑧 → ({𝑓} × 𝒫 𝑓) = ({𝑧} × 𝒫 𝑧))
34	33	cbviunv 4997	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓) = ∪ 𝑧 ∈ 𝑥 ({𝑧} × 𝒫 𝑧)
35		iuneq1 4968	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ∪ 𝑧 ∈ 𝑥 ({𝑧} × 𝒫 𝑧) = ∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧))
36	34, 35	eqtrid 2808	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓) = ∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧))
37	36	fveq2d 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓)) = (card‘∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧)))
38	37	cbvmptv 5209	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓))) = (𝑦 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑧 ∈ 𝑦 ({𝑧} × 𝒫 𝑧)))
39	38	ackbij1 10296	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑓 ∈ 𝑥 ({𝑓} × 𝒫 𝑓))):(𝒫 ω ∩ Fin)–1-1-onto→ω
40		f1oeng 8981	. . . . . . . . . . . . . 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 598	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ω ∩ Fin) ≈ ω)
42		domentr 9024	. . . . . . . . . . . . 13 ⊢ (((𝒫 ran 𝑓 ∩ Fin) ≼ (𝒫 ω ∩ Fin) ∧ (𝒫 ω ∩ Fin) ≈ ω) → (𝒫 ran 𝑓 ∩ Fin) ≼ ω)
43	30, 41, 42	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 ran 𝑓 ∩ Fin) ≼ ω)
44		endomtr 9023	. . . . . . . . . . . 12 ⊢ (((𝒫 𝐴 ∩ Fin) ≈ (𝒫 ran 𝑓 ∩ Fin) ∧ (𝒫 ran 𝑓 ∩ Fin) ≼ ω) → (𝒫 𝐴 ∩ Fin) ≼ ω)
45	19, 43, 44	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝒫 𝐴 ∩ Fin) ≼ ω)
46		domtr 9018	. . . . . . . . . . 11 ⊢ ((((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ (𝒫 𝐴 ∩ Fin) ∧ (𝒫 𝐴 ∩ Fin) ≼ ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω)
47	13, 45, 46	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω)
48		ondomen 10097	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card)
49	6, 47, 48	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∈ dom card)
50		eqid 2761	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦) = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦)
51	50	fifo 9408	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
52	51	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦):((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴))
53		fodomnum 10117	. . . . . . . . 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 66	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅}))
55		domtr 9018	. . . . . . . 8 ⊢ (((fi‘𝐴) ≼ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ∧ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ≼ ω) → (fi‘𝐴) ≼ ω)
56	54, 47, 55	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ 𝐵 ∧ ω ∈ V) ∧ 𝑓:𝐴–1-1→ω) → (fi‘𝐴) ≼ ω)
57	56	ex 418	. . . . . 6 ⊢ ((𝐴 ∈ 𝐵 ∧ ω ∈ V) → (𝑓:𝐴–1-1→ω → (fi‘𝐴) ≼ ω))
58	57	exlimdv 1966	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ω ∈ V) → (∃𝑓 𝑓:𝐴–1-1→ω → (fi‘𝐴) ≼ ω))
59	4, 58	sylan2 605	. . . 4 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≼ ω) → (∃𝑓 𝑓:𝐴–1-1→ω → (fi‘𝐴) ≼ ω))
60	2, 59	mpd 16	. . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ≼ ω) → (fi‘𝐴) ≼ ω)
61	60	ex 418	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ≼ ω → (fi‘𝐴) ≼ ω))
62		fvex 6890	. . . 4 ⊢ (fi‘𝐴) ∈ V
63		ssfii 9395	. . . 4 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ (fi‘𝐴))
64		ssdomg 9011	. . . 4 ⊢ ((fi‘𝐴) ∈ V → (𝐴 ⊆ (fi‘𝐴) → 𝐴 ≼ (fi‘𝐴)))
65	62, 63, 64	mpsyl 69	. . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ≼ (fi‘𝐴))
66		domtr 9018	. . . 4 ⊢ ((𝐴 ≼ (fi‘𝐴) ∧ (fi‘𝐴) ≼ ω) → 𝐴 ≼ ω)
67	66	ex 418	. . 3 ⊢ (𝐴 ≼ (fi‘𝐴) → ((fi‘𝐴) ≼ ω → 𝐴 ≼ ω))
68	65, 67	syl 18	. 2 ⊢ (𝐴 ∈ 𝐵 → ((fi‘𝐴) ≼ ω → 𝐴 ≼ ω))
69	61, 68	impbid 215	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ≼ ω ↔ (fi‘𝐴) ≼ ω))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∃wex 1812   ∈ wcel 2145  Vcvv 3451   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ∩ cint 4907  ∪ ciun 4951   class class class wbr 5103   ↦ cmpt 5186   × cxp 5649  dom cdm 5651  ran crn 5652   “ cima 5654  Oncon0 6355  –1-1→wf1 6528  –onto→wfo 6529  –1-1-onto→wf1o 6530  ‘cfv 6531  ωcom 7866   ≈ cen 8954   ≼ cdom 8955  Fincfn 8957  ficfi 9386  cardccrd 9997

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-oadd 8464  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fi 9387  df-dju 9963  df-card 10001  df-acn 10004

This theorem is used by:  2ndcsb  23747