Theorem ackbij1b 9732

Description: The Ackermann bijection, part 1b: the bijection from ackbij1 9731 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 9729	. . . . 5 ⊢ 𝐹:(𝒫 ω ∩ Fin)–1-1→ω
3		ackbij2lem1 9712	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ (𝒫 ω ∩ Fin))
4		pwexg 5242	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ∈ V)
5		f1imaeng 8608	. . . . 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 1467	. . . 4 ⊢ (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴)
7		nnfi 8759	. . . . . 6 ⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
8		pwfi 8769	. . . . . 6 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
9	7, 8	sylib 221	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ∈ Fin)
10		ficardid 9457	. . . . 5 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ≈ 𝒫 𝐴)
11		ensym 8597	. . . . 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 8600	. . . 4 ⊢ (((𝐹 “ 𝒫 𝐴) ≈ 𝒫 𝐴 ∧ 𝒫 𝐴 ≈ (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
14	6, 12, 13	syl2anc 587	. . 3 ⊢ (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ≈ (card‘𝒫 𝐴))
15		onfin2 8783	. . . . . . 7 ⊢ ω = (On ∩ Fin)
16		inss2 4118	. . . . . . 7 ⊢ (On ∩ Fin) ⊆ Fin
17	15, 16	eqsstri 3909	. . . . . 6 ⊢ ω ⊆ Fin
18		ficardom 9456	. . . . . . 7 ⊢ (𝒫 𝐴 ∈ Fin → (card‘𝒫 𝐴) ∈ ω)
19	9, 18	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ ω)
20	17, 19	sseldi 3873	. . . . 5 ⊢ (𝐴 ∈ ω → (card‘𝒫 𝐴) ∈ Fin)
21		php3 8746	. . . . . 6 ⊢ (((card‘𝒫 𝐴) ∈ Fin ∧ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴)) → (𝐹 “ 𝒫 𝐴) ≺ (card‘𝒫 𝐴))
22	21	ex 416	. . . . 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 8577	. . . 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 138	. 2 ⊢ (𝐴 ∈ ω → ¬ (𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴))
27		fvex 6681	. . . . . 6 ⊢ (𝐹‘𝑎) ∈ V
28		ackbij1lem3 9715	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → 𝐴 ∈ (𝒫 ω ∩ Fin))
29		elpwi 4494	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
30	1	ackbij1lem12 9724	. . . . . . . . 9 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝑎 ⊆ 𝐴) → (𝐹‘𝑎) ⊆ (𝐹‘𝐴))
31	28, 29, 30	syl2an 599	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → (𝐹‘𝑎) ⊆ (𝐹‘𝐴))
32	1	ackbij1lem10 9722	. . . . . . . . . . 11 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
33		peano1 7614	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
34	32, 33	f0cli 6868	. . . . . . . . . 10 ⊢ (𝐹‘𝑎) ∈ ω
35		nnord 7601	. . . . . . . . . 10 ⊢ ((𝐹‘𝑎) ∈ ω → Ord (𝐹‘𝑎))
36	34, 35	ax-mp 5	. . . . . . . . 9 ⊢ Ord (𝐹‘𝑎)
37	32, 33	f0cli 6868	. . . . . . . . . 10 ⊢ (𝐹‘𝐴) ∈ ω
38		nnord 7601	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ∈ ω → Ord (𝐹‘𝐴))
39	37, 38	ax-mp 5	. . . . . . . . 9 ⊢ Ord (𝐹‘𝐴)
40		ordsucsssuc 7551	. . . . . . . . 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 221	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹‘𝑎) ⊆ suc (𝐹‘𝐴))
43	1	ackbij1lem14 9726	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = suc (𝐹‘𝐴))
44	1	ackbij1lem8 9720	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐹‘{𝐴}) = (card‘𝒫 𝐴))
45	43, 44	eqtr3d 2775	. . . . . . . 8 ⊢ (𝐴 ∈ ω → suc (𝐹‘𝐴) = (card‘𝒫 𝐴))
46	45	adantr 484	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹‘𝐴) = (card‘𝒫 𝐴))
47	42, 46	sseqtrd 3915	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑎 ∈ 𝒫 𝐴) → suc (𝐹‘𝑎) ⊆ (card‘𝒫 𝐴))
48		sucssel 6258	. . . . . 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 3096	. . . 4 ⊢ (𝐴 ∈ ω → ∀𝑎 ∈ 𝒫 𝐴(𝐹‘𝑎) ∈ (card‘𝒫 𝐴))
51		f1fun 6570	. . . . . 6 ⊢ (𝐹:(𝒫 ω ∩ Fin)–1-1→ω → Fun 𝐹)
52	2, 51	ax-mp 5	. . . . 5 ⊢ Fun 𝐹
53		f1dm 6572	. . . . . . 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 3926	. . . . 5 ⊢ (𝐴 ∈ ω → 𝒫 𝐴 ⊆ dom 𝐹)
56		funimass4 6728	. . . . 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 590	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ∀𝑎 ∈ 𝒫 𝐴(𝐹‘𝑎) ∈ (card‘𝒫 𝐴)))
58	50, 57	mpbird 260	. . 3 ⊢ (𝐴 ∈ ω → (𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴))
59		sspss 3988	. . 3 ⊢ ((𝐹 “ 𝒫 𝐴) ⊆ (card‘𝒫 𝐴) ↔ ((𝐹 “ 𝒫 𝐴) ⊊ (card‘𝒫 𝐴) ∨ (𝐹 “ 𝒫 𝐴) = (card‘𝒫 𝐴)))
60	58, 59	sylib 221	. 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 209   ∧ wa 399   ∨ wo 846   = wceq 1542   ∈ wcel 2113  ∀wral 3053  Vcvv 3397   ∩ cin 3840   ⊆ wss 3841   ⊊ wpss 3842  𝒫 cpw 4485  {csn 4513  ∪ ciun 4878   class class class wbr 5027   ↦ cmpt 5107   × cxp 5517  dom cdm 5519   “ cima 5522  Ord word 6165  Oncon0 6166  suc csuc 6168  Fun wfun 6327  –1-1→wf1 6330  ‘cfv 6333  ωcom 7593   ≈ cen 8545   ≺ csdm 8547  Fincfn 8548  cardccrd 9430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-2o 8125  df-oadd 8128  df-er 8313  df-map 8432  df-en 8549  df-dom 8550  df-sdom 8551  df-fin 8552  df-dju 9396  df-card 9434

This theorem is referenced by:  ackbij2lem2  9733