Theorem cnfcom2 9697

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 7442	. . . . . . . . . 10 ⊢ (𝐹 supp ∅) ∈ V
11	5	oion 9531	. . . . . . . . . 10 ⊢ ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On)
12	10, 11	ax-mp 5	. . . . . . . . 9 ⊢ dom 𝐺 ∈ On
13	12	elexi 3494	. . . . . . . 8 ⊢ dom 𝐺 ∈ V
14	13	uniex 7731	. . . . . . 7 ⊢ ∪ dom 𝐺 ∈ V
15	14	sucid 6447	. . . . . 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 9696	. . . . . 6 ⊢ (𝜑 → dom 𝐺 = suc ∪ dom 𝐺)
19	15, 18	eleqtrrid 2841	. . . . 5 ⊢ (𝜑 → ∪ dom 𝐺 ∈ dom 𝐺)
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 19	cnfcom 9695	. . . 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 7420	. . . . . 6 ⊢ (ω ↑o 𝑊) = (ω ↑o (𝐺‘∪ dom 𝐺))
22	16	fveq2i 6895	. . . . . 6 ⊢ (𝐹‘𝑊) = (𝐹‘(𝐺‘∪ dom 𝐺))
23	21, 22	oveq12i 7421	. . . . 5 ⊢ ((ω ↑o 𝑊) ·o (𝐹‘𝑊)) = ((ω ↑o (𝐺‘∪ dom 𝐺)) ·o (𝐹‘(𝐺‘∪ dom 𝐺)))
24		f1oeq3 6824	. . . . 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 6896	. . . 4 ⊢ (𝜑 → (𝑇‘dom 𝐺) = (𝑇‘suc ∪ dom 𝐺))
28	27	f1oeq1d 6829	. . 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 257	. 2 ⊢ (𝜑 → (𝑇‘dom 𝐺):(𝐻‘suc ∪ dom 𝐺)–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊)))
30		omelon 9641	. . . . . . 7 ⊢ ω ∈ On
31	30	a1i 11	. . . . . 6 ⊢ (𝜑 → ω ∈ On)
32	1, 31, 2	cantnff1o 9691	. . . . . . . . 9 ⊢ (𝜑 → (ω CNF 𝐴):𝑆–1-1-onto→(ω ↑o 𝐴))
33		f1ocnv 6846	. . . . . . . . 9 ⊢ ((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑o 𝐴) → ◡(ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto→𝑆)
34		f1of 6834	. . . . . . . . 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 7088	. . . . . . 7 ⊢ (𝜑 → (◡(ω CNF 𝐴)‘𝐵) ∈ 𝑆)
37	4, 36	eqeltrid 2838	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
38	8	oveq1i 7419	. . . . . . . . . 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 7487	. . . . . . . 8 ⊢ (𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧))
41		eqid 2733	. . . . . . . 8 ⊢ ∅ = ∅
42		seqomeq12 8454	. . . . . . . 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 691	. . . . . . 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 2761	. . . . . 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 9663	. . . . 5 ⊢ (𝜑 → ((ω CNF 𝐴)‘𝐹) = (𝐻‘dom 𝐺))
46	4	fveq2i 6895	. . . . 5 ⊢ ((ω CNF 𝐴)‘𝐹) = ((ω CNF 𝐴)‘(◡(ω CNF 𝐴)‘𝐵))
47	45, 46	eqtr3di 2788	. . . 4 ⊢ (𝜑 → (𝐻‘dom 𝐺) = ((ω CNF 𝐴)‘(◡(ω CNF 𝐴)‘𝐵)))
48	18	fveq2d 6896	. . . 4 ⊢ (𝜑 → (𝐻‘dom 𝐺) = (𝐻‘suc ∪ dom 𝐺))
49		f1ocnvfv2 7275	. . . . 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 585	. . . 4 ⊢ (𝜑 → ((ω CNF 𝐴)‘(◡(ω CNF 𝐴)‘𝐵)) = 𝐵)
51	47, 48, 50	3eqtr3d 2781	. . 3 ⊢ (𝜑 → (𝐻‘suc ∪ dom 𝐺) = 𝐵)
52	51	f1oeq2d 6830	. 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 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∪ cun 3947  ∅c0 4323  ∪ cuni 4909   ↦ cmpt 5232   E cep 5580  ^◡ccnv 5676  dom cdm 5677  Oncon0 6365  suc csuc 6367  ⟶wf 6540  –1-1-onto→wf1o 6543  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411  ωcom 7855   supp csupp 8146  seq_ωcseqom 8447   +_o coa 8463   ·_o comu 8464   ↑_o coe 8465  OrdIsocoi 9504   CNF ccnf 9656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-seqom 8448  df-1o 8466  df-2o 8467  df-oadd 8470  df-omul 8471  df-oexp 8472  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-oi 9505  df-cnf 9657

This theorem is referenced by:  cnfcom3  9699