Theorem ackbij1b 10191

Description: The Ackermann bijection, part 1b: the bijection from ackbij1 10190 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 10188	. . . . 5 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3		ackbij2lem1 10171	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin))
4		pwexg 5333	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ∈ V)
5		f1imaeng 8985	. . . . 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 9131	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
8		pwfi 9268	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
9	7, 8	sylib 218	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ∈ Fin)
10		ficardid 9915	. . . . 5 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
11		ensym 8974	. . . . 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 8977	. . . 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 9180	. . . . . . 7 ⊢ ω = (On ∩ Fin)
16		inss2 4201	. . . . . . 7 ⊢ (On ∩ Fin) ⊆ Fin
17	15, 16	eqsstri 3993	. . . . . 6 ⊢ ω ⊆ Fin
18		ficardom 9914	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ∈ ω)
19	9, 18	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ ω)
20	17, 19	sselid 3944	. . . . 5 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ Fin)
21		php3 9173	. . . . . 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 8952	. . . 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 6871	. . . . . 6 ⊢ (𝐹‘𝑎) ∈ V
28		ackbij1lem3 10174	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝐴 ∈ (𝒫 ω ∩ Fin))
29		elpwi 4570	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
30	1	ackbij1lem12 10183	. . . . . . . . 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 10181	. . . . . . . . . . 11 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
33		peano1 7865	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
34	32, 33	f0cli 7070	. . . . . . . . . 10 ⊢ (𝐹‘𝑎) ∈ ω
35		nnord 7850	. . . . . . . . . 10 ⊢ ((𝐹‘𝑎) ∈ ω → Ord (𝐹‘𝑎))
36	34, 35	ax-mp 5	. . . . . . . . 9 ⊢ Ord (𝐹‘𝑎)
37	32, 33	f0cli 7070	. . . . . . . . . 10 ⊢ (𝐹‘𝐴) ∈ ω
38		nnord 7850	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ∈ ω → Ord (𝐹‘𝐴))
39	37, 38	ax-mp 5	. . . . . . . . 9 ⊢ Ord (𝐹‘𝐴)
40		ordsucsssuc 7798	. . . . . . . . 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 10185	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹‘𝐴))
44	1	ackbij1lem8 10179	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
45	43, 44	eqtr3d 2766	. . . . . . . 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 3983	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹‘𝑎) ⊆ (card‘𝒫 𝐴))
48		sucssel 6429	. . . . . 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 3125	. . . 4 ⊢ (𝐴 ∈ ω → ∀𝑎 ∈ 𝒫 𝐴(𝐹‘𝑎) ∈ (card‘𝒫 𝐴))
51		f1fun 6758	. . . . . 6 ⊢ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → Fun 𝐹)
52	2, 51	ax-mp 5	. . . . 5 ⊢ Fun 𝐹
53		f1dm 6760	. . . . . . 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 3988	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ dom 𝐹)
56		funimass4 6925	. . . . 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 4065	. . 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 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914   ⊊ wpss 3915  𝒫 cpw 4563  {csn 4589  ∪ ciun 4955   class class class wbr 5107   ↦ cmpt 5188   × cxp 5636  dom cdm 5638   “ cima 5641  Ord word 6331  Oncon0 6332  suc csuc 6334  Fun wfun 6505  –1-1→wf1 6508  ‘cfv 6511  ωcom 7842   ≈ cen 8915   ≺ csdm 8917  Fincfn 8918  cardccrd 9888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-dju 9854  df-card 9892

This theorem is referenced by:  ackbij2lem2  10192