Theorem ackbij1b 10129

Description: The Ackermann bijection, part 1b: the bijection from ackbij1 10128 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 10126	. . . . 5 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3		ackbij2lem1 10109	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin))
4		pwexg 5314	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ∈ V)
5		f1imaeng 8936	. . . . 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 1468	. . . 4 ⊢ (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴)
7		nnfi 9077	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
8		pwfi 9203	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
9	7, 8	sylib 218	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ∈ Fin)
10		ficardid 9855	. . . . 5 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
11		ensym 8925	. . . . 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 8928	. . . 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 9125	. . . . . . 7 ⊢ ω = (On ∩ Fin)
16		inss2 4185	. . . . . . 7 ⊢ (On ∩ Fin) ⊆ Fin
17	15, 16	eqsstri 3976	. . . . . 6 ⊢ ω ⊆ Fin
18		ficardom 9854	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ∈ ω)
19	9, 18	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ ω)
20	17, 19	sselid 3927	. . . . 5 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ Fin)
21		php3 9118	. . . . . 6 ⊢ (((card‘𝒫 𝐴) ∈ Fin ∧ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴))
22	21	ex 412	. . . . 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 8903	. . . 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 6835	. . . . . 6 ⊢ (𝐹‘𝑎) ∈ V
28		ackbij1lem3 10112	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝐴 ∈ (𝒫 ω ∩ Fin))
29		elpwi 4554	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
30	1	ackbij1lem12 10121	. . . . . . . . 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 10119	. . . . . . . . . . 11 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
33		peano1 7819	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
34	32, 33	f0cli 7031	. . . . . . . . . 10 ⊢ (𝐹‘𝑎) ∈ ω
35		nnord 7804	. . . . . . . . . 10 ⊢ ((𝐹‘𝑎) ∈ ω → Ord (𝐹‘𝑎))
36	34, 35	ax-mp 5	. . . . . . . . 9 ⊢ Ord (𝐹‘𝑎)
37	32, 33	f0cli 7031	. . . . . . . . . 10 ⊢ (𝐹‘𝐴) ∈ ω
38		nnord 7804	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ∈ ω → Ord (𝐹‘𝐴))
39	37, 38	ax-mp 5	. . . . . . . . 9 ⊢ Ord (𝐹‘𝐴)
40		ordsucsssuc 7753	. . . . . . . . 9 ⊢ ((Ord (𝐹‘𝑎) ∧ Ord (𝐹‘𝐴)) → ((𝐹‘𝑎) ⊆ (𝐹‘𝐴) ↔ suc (𝐹‘𝑎) ⊆ suc (𝐹‘𝐴)))
41	36, 39, 40	mp2an 692	. . . . . . . 8 ⊢ ((𝐹‘𝑎) ⊆ (𝐹‘𝐴) ↔ suc (𝐹‘𝑎) ⊆ suc (𝐹‘𝐴))
42	31, 41	sylib 218	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹‘𝑎) ⊆ suc (𝐹‘𝐴))
43	1	ackbij1lem14 10123	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹‘𝐴))
44	1	ackbij1lem8 10117	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
45	43, 44	eqtr3d 2768	. . . . . . . 8 ⊢ (𝐴 ∈ ω → suc (𝐹‘𝐴) = (card‘𝒫 𝐴))
46	45	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹‘𝐴) = (card‘𝒫 𝐴))
47	42, 46	sseqtrd 3966	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹‘𝑎) ⊆ (card‘𝒫 𝐴))
48		sucssel 6403	. . . . . 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 3124	. . . 4 ⊢ (𝐴 ∈ ω → ∀𝑎 ∈ 𝒫 𝐴(𝐹‘𝑎) ∈ (card‘𝒫 𝐴))
51		f1fun 6721	. . . . . 6 ⊢ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → Fun 𝐹)
52	2, 51	ax-mp 5	. . . . 5 ⊢ Fun 𝐹
53		f1dm 6723	. . . . . . 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 3971	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ dom 𝐹)
56		funimass4 6886	. . . . 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 257	. . 3 ⊢ (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴))
59		sspss 4049	. . 3 ⊢ ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) ∨ (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)))
60	58, 59	sylib 218	. 2 ⊢ (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) ∨ (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)))
61		orel1 888	. 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 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ∩ cin 3896   ⊆ wss 3897   ⊊ wpss 3898  𝒫 cpw 4547  {csn 4573  ∪ ciun 4939   class class class wbr 5089   ↦ cmpt 5170   × cxp 5612  dom cdm 5614   “ cima 5617  Ord word 6305  Oncon0 6306  suc csuc 6308  Fun wfun 6475  –1-1→wf1 6478  ‘cfv 6481  ωcom 7796   ≈ cen 8866   ≺ csdm 8868  Fincfn 8869  cardccrd 9828

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-dju 9794  df-card 9832

This theorem is referenced by:  ackbij2lem2  10130