Theorem ackbij1b 10216

Description: The Ackermann bijection, part 1b: the bijection from ackbij1 10215 restricts naturally to the powers of particular naturals. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Hypothesis

Ref	Expression
ackbij.f	⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))

Assertion

Ref	Expression
ackbij1b	⊢ (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦

Proof of Theorem ackbij1b

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ackbij.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
2	1	ackbij1lem17 10213	. . . . 5 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3		ackbij2lem1 10196	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin))
4		pwexg 5369	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ∈ V)
5		f1imaeng 8993	. . . . 5 ⊢ ((𝐹:(𝒫 ω ∩ Fin)–1-1→ω ∧ 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin) ∧ 𝒫 𝐴 ∈ V) → (𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴)
6	2, 3, 4, 5	mp3an2i 1466	. . . 4 ⊢ (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴)
7		nnfi 9150	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
8		pwfi 9161	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
9	7, 8	sylib 217	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ∈ Fin)
10		ficardid 9939	. . . . 5 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
11		ensym 8982	. . . . 5 ⊢ ((card‘𝒫 𝐴) ≈ 𝒫 𝐴 → 𝒫 𝐴 ≈ (card‘𝒫 𝐴))
12	9, 10, 11	3syl 18	. . . 4 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ≈ (card‘𝒫 𝐴))
13		entr 8985	. . . 4 ⊢ (((𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴 ∧ 𝒫 𝐴 ≈ (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
14	6, 12, 13	syl2anc 584	. . 3 ⊢ (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
15		onfin2 9214	. . . . . . 7 ⊢ ω = (On ∩ Fin)
16		inss2 4225	. . . . . . 7 ⊢ (On ∩ Fin) ⊆ Fin
17	15, 16	eqsstri 4012	. . . . . 6 ⊢ ω ⊆ Fin
18		ficardom 9938	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ∈ ω)
19	9, 18	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ ω)
20	17, 19	sselid 3976	. . . . 5 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ Fin)
21		php3 9195	. . . . . 6 ⊢ (((card‘𝒫 𝐴) ∈ Fin ∧ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴))
22	21	ex 413	. . . . 5 ⊢ ((card‘𝒫 𝐴) ∈ Fin → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) → (𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴)))
23	20, 22	syl 17	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) → (𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴)))
24		sdomnen 8960	. . . 4 ⊢ ((𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴) → ¬ (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
25	23, 24	syl6 35	. . 3 ⊢ (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) → ¬ (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴)))
26	14, 25	mt2d 136	. 2 ⊢ (𝐴 ∈ ω → ¬ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴))
27		fvex 6891	. . . . . 6 ⊢ (𝐹‘𝑎) ∈ V
28		ackbij1lem3 10199	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝐴 ∈ (𝒫 ω ∩ Fin))
29		elpwi 4603	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
30	1	ackbij1lem12 10208	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ⊆ 𝐴) → (𝐹‘𝑎) ⊆ (𝐹‘𝐴))
31	28, 29, 30	syl2an 596	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → (𝐹‘𝑎) ⊆ (𝐹‘𝐴))
32	1	ackbij1lem10 10206	. . . . . . . . . . 11 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
33		peano1 7861	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
34	32, 33	f0cli 7084	. . . . . . . . . 10 ⊢ (𝐹‘𝑎) ∈ ω
35		nnord 7846	. . . . . . . . . 10 ⊢ ((𝐹‘𝑎) ∈ ω → Ord (𝐹‘𝑎))
36	34, 35	ax-mp 5	. . . . . . . . 9 ⊢ Ord (𝐹‘𝑎)
37	32, 33	f0cli 7084	. . . . . . . . . 10 ⊢ (𝐹‘𝐴) ∈ ω
38		nnord 7846	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ∈ ω → Ord (𝐹‘𝐴))
39	37, 38	ax-mp 5	. . . . . . . . 9 ⊢ Ord (𝐹‘𝐴)
40		ordsucsssuc 7794	. . . . . . . . 9 ⊢ ((Ord (𝐹‘𝑎) ∧ Ord (𝐹‘𝐴)) → ((𝐹‘𝑎) ⊆ (𝐹‘𝐴) ↔ suc (𝐹‘𝑎) ⊆ suc (𝐹‘𝐴)))
41	36, 39, 40	mp2an 690	. . . . . . . 8 ⊢ ((𝐹‘𝑎) ⊆ (𝐹‘𝐴) ↔ suc (𝐹‘𝑎) ⊆ suc (𝐹‘𝐴))
42	31, 41	sylib 217	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹‘𝑎) ⊆ suc (𝐹‘𝐴))
43	1	ackbij1lem14 10210	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹‘𝐴))
44	1	ackbij1lem8 10204	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
45	43, 44	eqtr3d 2773	. . . . . . . 8 ⊢ (𝐴 ∈ ω → suc (𝐹‘𝐴) = (card‘𝒫 𝐴))
46	45	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹‘𝐴) = (card‘𝒫 𝐴))
47	42, 46	sseqtrd 4018	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹‘𝑎) ⊆ (card‘𝒫 𝐴))
48		sucssel 6448	. . . . . 6 ⊢ ((𝐹‘𝑎) ∈ V → (suc (𝐹‘𝑎) ⊆ (card‘𝒫 𝐴) → (𝐹‘𝑎) ∈ (card‘𝒫 𝐴)))
49	27, 47, 48	mpsyl 68	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → (𝐹‘𝑎) ∈ (card‘𝒫 𝐴))
50	49	ralrimiva 3145	. . . 4 ⊢ (𝐴 ∈ ω → ∀𝑎 ∈ 𝒫 𝐴(𝐹‘𝑎) ∈ (card‘𝒫 𝐴))
51		f1fun 6776	. . . . . 6 ⊢ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → Fun 𝐹)
52	2, 51	ax-mp 5	. . . . 5 ⊢ Fun 𝐹
53		f1dm 6778	. . . . . . 7 ⊢ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → dom 𝐹 = (𝒫 ω ∩ Fin))
54	2, 53	ax-mp 5	. . . . . 6 ⊢ dom 𝐹 = (𝒫 ω ∩ Fin)
55	3, 54	sseqtrrdi 4029	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ dom 𝐹)
56		funimass4 6943	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝒫 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ∀𝑎 ∈ 𝒫 𝐴(𝐹‘𝑎) ∈ (card‘𝒫 𝐴)))
57	52, 55, 56	sylancr 587	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ∀𝑎 ∈ 𝒫 𝐴(𝐹‘𝑎) ∈ (card‘𝒫 𝐴)))
58	50, 57	mpbird 256	. . 3 ⊢ (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴))
59		sspss 4095	. . 3 ⊢ ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) ∨ (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)))
60	58, 59	sylib 217	. 2 ⊢ (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) ∨ (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)))
61		orel1 887	. 2 ⊢ (¬ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) → (((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) ∨ (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)))
62	26, 60, 61	sylc 65	1 ⊢ (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  ∀wral 3060  Vcvv 3473   ∩ cin 3943   ⊆ wss 3944   ⊊ wpss 3945  𝒫 cpw 4596  {csn 4622  ∪ ciun 4990   class class class wbr 5141   ↦ cmpt 5224   × cxp 5667  dom cdm 5669   “ cima 5672  Ord word 6352  Oncon0 6353  suc csuc 6355  Fun wfun 6526  –1-1→wf1 6529  ‘cfv 6532  ωcom 7838   ≈ cen 8919   ≺ csdm 8921  Fincfn 8922  cardccrd 9912

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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-2o 8449  df-oadd 8452  df-er 8686  df-map 8805  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-dju 9878  df-card 9916

This theorem is referenced by:  ackbij2lem2  10217