Theorem cnfcom2 9696

Description: Any nonzero ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)

Hypotheses

Ref	Expression
cnfcom.s	⊢ 𝑆 = dom (ω CNF 𝐴)
cnfcom.a	⊢ (𝜑 → 𝐴 ∈ On)
cnfcom.b	⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))
cnfcom.f	⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)
cnfcom.g	⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
cnfcom.h	⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)
cnfcom.t	⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
cnfcom.m	⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))
cnfcom.k	⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))
cnfcom.w	⊢ 𝑊 = (𝐺‘∪ dom 𝐺)
cnfcom2.1	⊢ (𝜑 → ∅ ∈ 𝐵)

Assertion

Ref	Expression
cnfcom2	⊢ (𝜑 → (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊)))

Distinct variable groups:   𝑥,𝑘,𝑧,𝐴   𝑥,𝑀   𝑓,𝑘,𝑥,𝑧,𝐹   𝑧,𝑇   𝑥,𝑊   𝑓,𝐺,𝑘,𝑥,𝑧   𝑓,𝐻,𝑥   𝑆,𝑘,𝑧   𝜑,𝑘,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑓)   𝐴(𝑓)   𝐵(𝑥,𝑧,𝑓,𝑘)   𝑆(𝑥,𝑓)   𝑇(𝑥,𝑓,𝑘)   𝐻(𝑧,𝑘)   𝐾(𝑥,𝑧,𝑓,𝑘)   𝑀(𝑧,𝑓,𝑘)   𝑊(𝑧,𝑓,𝑘)

Proof of Theorem cnfcom2

Step	Hyp	Ref	Expression
1		cnfcom.s	. . . . 5 ⊢ 𝑆 = dom (ω CNF 𝐴)
2		cnfcom.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ On)
3		cnfcom.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))
4		cnfcom.f	. . . . 5 ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)
5		cnfcom.g	. . . . 5 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
6		cnfcom.h	. . . . 5 ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)
7		cnfcom.t	. . . . 5 ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
8		cnfcom.m	. . . . 5 ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))
9		cnfcom.k	. . . . 5 ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))
10		ovex 7441	. . . . . . . . . 10 ⊢ (𝐹 supp ∅) ∈ V
11	5	oion 9530	. . . . . . . . . 10 ⊢ ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On)
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ dom 𝐺 ∈ On
13	12	elexi 3493	. . . . . . . 8 ⊢ dom 𝐺 ∈ V
14	13	uniex 7730	. . . . . . 7 ⊢ ∪ dom 𝐺 ∈ V
15	14	sucid 6446	. . . . . 6 ⊢ ∪ dom 𝐺 ∈ suc ∪ dom 𝐺
16		cnfcom.w	. . . . . . 7 ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)
17		cnfcom2.1	. . . . . . 7 ⊢ (𝜑 → ∅ ∈ 𝐵)
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17	cnfcom2lem 9695	. . . . . 6 ⊢ (𝜑 → dom 𝐺 = suc ∪ dom 𝐺)
19	15, 18	eleqtrrid 2840	. . . . 5 ⊢ (𝜑 → ∪ dom 𝐺 ∈ dom 𝐺)
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 19	cnfcom 9694	. . . 4 ⊢ (𝜑 → (𝑇‘suc ∪ dom 𝐺):(𝐻‘suc ∪ dom 𝐺)–1-1-onto→((ω ↑o (𝐺‘∪ dom 𝐺)) ·o (𝐹‘(𝐺‘∪ dom 𝐺))))
21	16	oveq2i 7419	. . . . . 6 ⊢ (ω ↑o 𝑊) = (ω ↑o (𝐺‘∪ dom 𝐺))
22	16	fveq2i 6894	. . . . . 6 ⊢ (𝐹‘𝑊) = (𝐹‘(𝐺‘∪ dom 𝐺))
23	21, 22	oveq12i 7420	. . . . 5 ⊢ ((ω ↑o 𝑊) ·o (𝐹‘𝑊)) = ((ω ↑o (𝐺‘∪ dom 𝐺)) ·o (𝐹‘(𝐺‘∪ dom 𝐺)))
24		f1oeq3 6823	. . . . 5 ⊢ (((ω ↑o 𝑊) ·o (𝐹‘𝑊)) = ((ω ↑o (𝐺‘∪ dom 𝐺)) ·o (𝐹‘(𝐺‘∪ dom 𝐺))) → ((𝑇‘suc ∪ dom 𝐺):(𝐻‘suc ∪ dom 𝐺)–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊)) ↔ (𝑇‘suc ∪ dom 𝐺):(𝐻‘suc ∪ dom 𝐺)–1-1-onto→((ω ↑o (𝐺‘∪ dom 𝐺)) ·o (𝐹‘(𝐺‘∪ dom 𝐺)))))
25	23, 24	ax-mp 5	. . . 4 ⊢ ((𝑇‘suc ∪ dom 𝐺):(𝐻‘suc ∪ dom 𝐺)–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊)) ↔ (𝑇‘suc ∪ dom 𝐺):(𝐻‘suc ∪ dom 𝐺)–1-1-onto→((ω ↑o (𝐺‘∪ dom 𝐺)) ·o (𝐹‘(𝐺‘∪ dom 𝐺))))
26	20, 25	sylibr 233	. . 3 ⊢ (𝜑 → (𝑇‘suc ∪ dom 𝐺):(𝐻‘suc ∪ dom 𝐺)–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊)))
27	18	fveq2d 6895	. . . 4 ⊢ (𝜑 → (𝑇‘dom 𝐺) = (𝑇‘suc ∪ dom 𝐺))
28	27	f1oeq1d 6828	. . 3 ⊢ (𝜑 → ((𝑇‘dom 𝐺):(𝐻‘suc ∪ dom 𝐺)–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊)) ↔ (𝑇‘suc ∪ dom 𝐺):(𝐻‘suc ∪ dom 𝐺)–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊))))
29	26, 28	mpbird 256	. 2 ⊢ (𝜑 → (𝑇‘dom 𝐺):(𝐻‘suc ∪ dom 𝐺)–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊)))
30		omelon 9640	. . . . . . 7 ⊢ ω ∈ On
31	30	a1i 11	. . . . . 6 ⊢ (𝜑 → ω ∈ On)
32	1, 31, 2	cantnff1o 9690	. . . . . . . . 9 ⊢ (𝜑 → (ω CNF 𝐴):𝑆–1-1-onto→(ω ↑o 𝐴))
33		f1ocnv 6845	. . . . . . . . 9 ⊢ ((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑o 𝐴) → ◡(ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto→𝑆)
34		f1of 6833	. . . . . . . . 9 ⊢ (◡(ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto→𝑆 → ◡(ω CNF 𝐴):(ω ↑o 𝐴)⟶𝑆)
35	32, 33, 34	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ◡(ω CNF 𝐴):(ω ↑o 𝐴)⟶𝑆)
36	35, 3	ffvelcdmd 7087	. . . . . . 7 ⊢ (𝜑 → (◡(ω CNF 𝐴)‘𝐵) ∈ 𝑆)
37	4, 36	eqeltrid 2837	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
38	8	oveq1i 7418	. . . . . . . . . 10 ⊢ (𝑀 +o 𝑧) = (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)
39	38	a1i 11	. . . . . . . . 9 ⊢ ((𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑀 +o 𝑧) = (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧))
40	39	mpoeq3ia 7486	. . . . . . . 8 ⊢ (𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧))
41		eqid 2732	. . . . . . . 8 ⊢ ∅ = ∅
42		seqomeq12 8453	. . . . . . . 8 ⊢ (((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)) ∧ ∅ = ∅) → seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅))
43	40, 41, 42	mp2an 690	. . . . . . 7 ⊢ seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)
44	6, 43	eqtri 2760	. . . . . 6 ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)
45	1, 31, 2, 5, 37, 44	cantnfval 9662	. . . . 5 ⊢ (𝜑 → ((ω CNF 𝐴)‘𝐹) = (𝐻‘dom 𝐺))
46	4	fveq2i 6894	. . . . 5 ⊢ ((ω CNF 𝐴)‘𝐹) = ((ω CNF 𝐴)‘(◡(ω CNF 𝐴)‘𝐵))
47	45, 46	eqtr3di 2787	. . . 4 ⊢ (𝜑 → (𝐻‘dom 𝐺) = ((ω CNF 𝐴)‘(◡(ω CNF 𝐴)‘𝐵)))
48	18	fveq2d 6895	. . . 4 ⊢ (𝜑 → (𝐻‘dom 𝐺) = (𝐻‘suc ∪ dom 𝐺))
49		f1ocnvfv2 7274	. . . . 5 ⊢ (((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑o 𝐴) ∧ 𝐵 ∈ (ω ↑o 𝐴)) → ((ω CNF 𝐴)‘(◡(ω CNF 𝐴)‘𝐵)) = 𝐵)
50	32, 3, 49	syl2anc 584	. . . 4 ⊢ (𝜑 → ((ω CNF 𝐴)‘(◡(ω CNF 𝐴)‘𝐵)) = 𝐵)
51	47, 48, 50	3eqtr3d 2780	. . 3 ⊢ (𝜑 → (𝐻‘suc ∪ dom 𝐺) = 𝐵)
52	51	f1oeq2d 6829	. 2 ⊢ (𝜑 → ((𝑇‘dom 𝐺):(𝐻‘suc ∪ dom 𝐺)–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊)) ↔ (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊))))
53	29, 52	mpbid 231	1 ⊢ (𝜑 → (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∪ cun 3946  ∅c0 4322  ∪ cuni 4908   ↦ cmpt 5231   E cep 5579  ^◡ccnv 5675  dom cdm 5676  Oncon0 6364  suc csuc 6366  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7408   ∈ cmpo 7410  ωcom 7854   supp csupp 8145  seq_ωcseqom 8446   +_o coa 8462   ·_o comu 8463   ↑_o coe 8464  OrdIsocoi 9503   CNF ccnf 9655

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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-inf2 9635

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-supp 8146  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-seqom 8447  df-1o 8465  df-2o 8466  df-oadd 8469  df-omul 8470  df-oexp 8471  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-fsupp 9361  df-oi 9504  df-cnf 9656

This theorem is referenced by:  cnfcom3  9698