Theorem cnfcom3 8484

Description: Any infinite ordinal 𝐵 is equinumerous to a power of ω. (We are being careful here to show explicit bijections rather than simple equinumerosity because we want a uniform construction for cnfcom3c 8486.) (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 4-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)
cnfcom3.1	⊢ (𝜑 → ω ⊆ 𝐵)
cnfcom.x	⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((𝐹‘𝑊) ·𝑜 𝑣) +𝑜 𝑢))
cnfcom.y	⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑢) +𝑜 𝑣))
cnfcom.n	⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺))

Assertion

Ref	Expression
cnfcom3	⊢ (𝜑 → 𝑁:𝐵–1-1-onto→(ω ↑𝑜 𝑊))

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

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

Proof of Theorem cnfcom3

Step	Hyp	Ref	Expression
1		omelon 8426	. . . . . 6 ⊢ ω ∈ On
2		cnfcom.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
3		suppssdm 7195	. . . . . . . . 9 ⊢ (𝐹 supp ∅) ⊆ dom 𝐹
4		cnfcom.f	. . . . . . . . . . . . 13 ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)
5		cnfcom.s	. . . . . . . . . . . . . . . 16 ⊢ 𝑆 = dom (ω CNF 𝐴)
6	1	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ω ∈ On)
7	5, 6, 2	cantnff1o 8476	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ω CNF 𝐴):𝑆–1-1-onto→(ω ↑𝑜 𝐴))
8		f1ocnv 6062	. . . . . . . . . . . . . . 15 ⊢ ((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑𝑜 𝐴) → ◡(ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto→𝑆)
9		f1of 6050	. . . . . . . . . . . . . . 15 ⊢ (◡(ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto→𝑆 → ◡(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
10	7, 8, 9	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ◡(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
11		cnfcom.b	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ (ω ↑𝑜 𝐴))
12	10, 11	ffvelrnd 6268	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡(ω CNF 𝐴)‘𝐵) ∈ 𝑆)
13	4, 12	syl5eqel 2692	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
14	5, 6, 2	cantnfs 8446	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
15	13, 14	mpbid 221	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
16	15	simpld 474	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶ω)
17		fdm 5964	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶ω → dom 𝐹 = 𝐴)
18	16, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
19	3, 18	syl5sseq 3616	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
20		cnfcom.w	. . . . . . . . 9 ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)
21		ovex 6577	. . . . . . . . . . . . . . 15 ⊢ (𝐹 supp ∅) ∈ V
22		cnfcom.g	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
23	22	oion 8324	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On)
24	21, 23	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom 𝐺 ∈ On
25	24	elexi 3186	. . . . . . . . . . . . 13 ⊢ dom 𝐺 ∈ V
26	25	uniex 6851	. . . . . . . . . . . 12 ⊢ ∪ dom 𝐺 ∈ V
27	26	sucid 5721	. . . . . . . . . . 11 ⊢ ∪ dom 𝐺 ∈ suc ∪ dom 𝐺
28		cnfcom.h	. . . . . . . . . . . 12 ⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)
29		cnfcom.t	. . . . . . . . . . . 12 ⊢ 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
30		cnfcom.m	. . . . . . . . . . . 12 ⊢ 𝑀 = ((ω ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘)))
31		cnfcom.k	. . . . . . . . . . . 12 ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
32		cnfcom3.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → ω ⊆ 𝐵)
33		peano1 6977	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ ω
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∅ ∈ ω)
35	32, 34	sseldd 3569	. . . . . . . . . . . 12 ⊢ (𝜑 → ∅ ∈ 𝐵)
36	5, 2, 11, 4, 22, 28, 29, 30, 31, 20, 35	cnfcom2lem 8481	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐺 = suc ∪ dom 𝐺)
37	27, 36	syl5eleqr 2695	. . . . . . . . . 10 ⊢ (𝜑 → ∪ dom 𝐺 ∈ dom 𝐺)
38	22	oif 8318	. . . . . . . . . . 11 ⊢ 𝐺:dom 𝐺⟶(𝐹 supp ∅)
39	38	ffvelrni 6266	. . . . . . . . . 10 ⊢ (∪ dom 𝐺 ∈ dom 𝐺 → (𝐺‘∪ dom 𝐺) ∈ (𝐹 supp ∅))
40	37, 39	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘∪ dom 𝐺) ∈ (𝐹 supp ∅))
41	20, 40	syl5eqel 2692	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ (𝐹 supp ∅))
42	19, 41	sseldd 3569	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐴)
43		onelon 5665	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑊 ∈ 𝐴) → 𝑊 ∈ On)
44	2, 42, 43	syl2anc 691	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ On)
45		oecl 7504	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑𝑜 𝑊) ∈ On)
46	1, 44, 45	sylancr 694	. . . . 5 ⊢ (𝜑 → (ω ↑𝑜 𝑊) ∈ On)
47	16, 42	ffvelrnd 6268	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑊) ∈ ω)
48		nnon 6963	. . . . . 6 ⊢ ((𝐹‘𝑊) ∈ ω → (𝐹‘𝑊) ∈ On)
49	47, 48	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑊) ∈ On)
50		cnfcom.y	. . . . . 6 ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑢) +𝑜 𝑣))
51		cnfcom.x	. . . . . 6 ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((𝐹‘𝑊) ·𝑜 𝑣) +𝑜 𝑢))
52	50, 51	omf1o 7948	. . . . 5 ⊢ (((ω ↑𝑜 𝑊) ∈ On ∧ (𝐹‘𝑊) ∈ On) → (𝑋 ∘ ◡𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹‘𝑊))–1-1-onto→((𝐹‘𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
53	46, 49, 52	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝑋 ∘ ◡𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹‘𝑊))–1-1-onto→((𝐹‘𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
54		ffn 5958	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶ω → 𝐹 Fn 𝐴)
55	16, 54	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn 𝐴)
56		0ex 4718	. . . . . . . . . . 11 ⊢ ∅ ∈ V
57	56	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ V)
58		elsuppfn 7190	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ On ∧ ∅ ∈ V) → (𝑊 ∈ (𝐹 supp ∅) ↔ (𝑊 ∈ 𝐴 ∧ (𝐹‘𝑊) ≠ ∅)))
59	55, 2, 57, 58	syl3anc 1318	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ∈ (𝐹 supp ∅) ↔ (𝑊 ∈ 𝐴 ∧ (𝐹‘𝑊) ≠ ∅)))
60		simpr 476	. . . . . . . . 9 ⊢ ((𝑊 ∈ 𝐴 ∧ (𝐹‘𝑊) ≠ ∅) → (𝐹‘𝑊) ≠ ∅)
61	59, 60	syl6bi 242	. . . . . . . 8 ⊢ (𝜑 → (𝑊 ∈ (𝐹 supp ∅) → (𝐹‘𝑊) ≠ ∅))
62	41, 61	mpd 15	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑊) ≠ ∅)
63		on0eln0 5697	. . . . . . . 8 ⊢ ((𝐹‘𝑊) ∈ On → (∅ ∈ (𝐹‘𝑊) ↔ (𝐹‘𝑊) ≠ ∅))
64	47, 48, 63	3syl 18	. . . . . . 7 ⊢ (𝜑 → (∅ ∈ (𝐹‘𝑊) ↔ (𝐹‘𝑊) ≠ ∅))
65	62, 64	mpbird 246	. . . . . 6 ⊢ (𝜑 → ∅ ∈ (𝐹‘𝑊))
66	5, 2, 11, 4, 22, 28, 29, 30, 31, 20, 32	cnfcom3lem 8483	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1𝑜))
67		ondif1 7468	. . . . . . . 8 ⊢ (𝑊 ∈ (On ∖ 1𝑜) ↔ (𝑊 ∈ On ∧ ∅ ∈ 𝑊))
68	67	simprbi 479	. . . . . . 7 ⊢ (𝑊 ∈ (On ∖ 1𝑜) → ∅ ∈ 𝑊)
69	66, 68	syl 17	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 𝑊)
70		omabs 7614	. . . . . 6 ⊢ ((((𝐹‘𝑊) ∈ ω ∧ ∅ ∈ (𝐹‘𝑊)) ∧ (𝑊 ∈ On ∧ ∅ ∈ 𝑊)) → ((𝐹‘𝑊) ·𝑜 (ω ↑𝑜 𝑊)) = (ω ↑𝑜 𝑊))
71	47, 65, 44, 69, 70	syl22anc 1319	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑊) ·𝑜 (ω ↑𝑜 𝑊)) = (ω ↑𝑜 𝑊))
72		f1oeq3 6042	. . . . 5 ⊢ (((𝐹‘𝑊) ·𝑜 (ω ↑𝑜 𝑊)) = (ω ↑𝑜 𝑊) → ((𝑋 ∘ ◡𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹‘𝑊))–1-1-onto→((𝐹‘𝑊) ·𝑜 (ω ↑𝑜 𝑊)) ↔ (𝑋 ∘ ◡𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹‘𝑊))–1-1-onto→(ω ↑𝑜 𝑊)))
73	71, 72	syl 17	. . . 4 ⊢ (𝜑 → ((𝑋 ∘ ◡𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹‘𝑊))–1-1-onto→((𝐹‘𝑊) ·𝑜 (ω ↑𝑜 𝑊)) ↔ (𝑋 ∘ ◡𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹‘𝑊))–1-1-onto→(ω ↑𝑜 𝑊)))
74	53, 73	mpbid 221	. . 3 ⊢ (𝜑 → (𝑋 ∘ ◡𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹‘𝑊))–1-1-onto→(ω ↑𝑜 𝑊))
75	5, 2, 11, 4, 22, 28, 29, 30, 31, 20, 35	cnfcom2 8482	. . 3 ⊢ (𝜑 → (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹‘𝑊)))
76		f1oco 6072	. . 3 ⊢ (((𝑋 ∘ ◡𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹‘𝑊))–1-1-onto→(ω ↑𝑜 𝑊) ∧ (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹‘𝑊))) → ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺)):𝐵–1-1-onto→(ω ↑𝑜 𝑊))
77	74, 75, 76	syl2anc 691	. 2 ⊢ (𝜑 → ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺)):𝐵–1-1-onto→(ω ↑𝑜 𝑊))
78		cnfcom.n	. . 3 ⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺))
79		f1oeq1 6040	. . 3 ⊢ (𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺)) → (𝑁:𝐵–1-1-onto→(ω ↑𝑜 𝑊) ↔ ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺)):𝐵–1-1-onto→(ω ↑𝑜 𝑊)))
80	78, 79	ax-mp 5	. 2 ⊢ (𝑁:𝐵–1-1-onto→(ω ↑𝑜 𝑊) ↔ ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺)):𝐵–1-1-onto→(ω ↑𝑜 𝑊))
81	77, 80	sylibr 223	1 ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→(ω ↑𝑜 𝑊))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643   E cep 4947  ^◡ccnv 5037  dom cdm 5038   ∘ ccom 5042  Oncon0 5640  suc csuc 5642   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   supp csupp 7182  seq_𝜔cseqom 7429  1_𝑜c1o 7440   +_𝑜 coa 7444   ·_𝑜 comu 7445   ↑_𝑜 coe 7446   finSupp cfsupp 8158  OrdIsocoi 8297   CNF ccnf 8441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-cnf 8442

This theorem is referenced by:  cnfcom3clem  8485