Theorem cnfcom3 8601

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 8603.) (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 8543	. . . . . 6 ⊢ ω ∈ On
2		cnfcom.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
3		suppssdm 7308	. . . . . . . . 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 8593	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ω CNF 𝐴):𝑆–1-1-onto→(ω ↑𝑜 𝐴))
8		f1ocnv 6149	. . . . . . . . . . . . . . 15 ⊢ ((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑𝑜 𝐴) → ◡(ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto→𝑆)
9		f1of 6137	. . . . . . . . . . . . . . 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 6360	. . . . . . . . . . . . 13 ⊢ (𝜑 → (◡(ω CNF 𝐴)‘𝐵) ∈ 𝑆)
13	4, 12	syl5eqel 2705	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
14	5, 6, 2	cantnfs 8563	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
15	13, 14	mpbid 222	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
16	15	simpld 475	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐴⟶ω)
17		fdm 6051	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶ω → dom 𝐹 = 𝐴)
18	16, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = 𝐴)
19	3, 18	syl5sseq 3653	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
20		cnfcom.w	. . . . . . . . 9 ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)
21		ovex 6678	. . . . . . . . . . . . . . 15 ⊢ (𝐹 supp ∅) ∈ V
22		cnfcom.g	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
23	22	oion 8441	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On)
24	21, 23	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom 𝐺 ∈ On
25	24	elexi 3213	. . . . . . . . . . . . 13 ⊢ dom 𝐺 ∈ V
26	25	uniex 6953	. . . . . . . . . . . 12 ⊢ ∪ dom 𝐺 ∈ V
27	26	sucid 5804	. . . . . . . . . . 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 7085	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ ω
34	33	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∅ ∈ ω)
35	32, 34	sseldd 3604	. . . . . . . . . . . 12 ⊢ (𝜑 → ∅ ∈ 𝐵)
36	5, 2, 11, 4, 22, 28, 29, 30, 31, 20, 35	cnfcom2lem 8598	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐺 = suc ∪ dom 𝐺)
37	27, 36	syl5eleqr 2708	. . . . . . . . . 10 ⊢ (𝜑 → ∪ dom 𝐺 ∈ dom 𝐺)
38	22	oif 8435	. . . . . . . . . . 11 ⊢ 𝐺:dom 𝐺⟶(𝐹 supp ∅)
39	38	ffvelrni 6358	. . . . . . . . . 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 2705	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ (𝐹 supp ∅))
42	19, 41	sseldd 3604	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐴)
43		onelon 5748	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑊 ∈ 𝐴) → 𝑊 ∈ On)
44	2, 42, 43	syl2anc 693	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ On)
45		oecl 7617	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑𝑜 𝑊) ∈ On)
46	1, 44, 45	sylancr 695	. . . . 5 ⊢ (𝜑 → (ω ↑𝑜 𝑊) ∈ On)
47	16, 42	ffvelrnd 6360	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑊) ∈ ω)
48		nnon 7071	. . . . . 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 8063	. . . . 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 693	. . . 4 ⊢ (𝜑 → (𝑋 ∘ ◡𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹‘𝑊))–1-1-onto→((𝐹‘𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
54		ffn 6045	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶ω → 𝐹 Fn 𝐴)
55	16, 54	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn 𝐴)
56		0ex 4790	. . . . . . . . . . 11 ⊢ ∅ ∈ V
57	56	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ V)
58		elsuppfn 7303	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ On ∧ ∅ ∈ V) → (𝑊 ∈ (𝐹 supp ∅) ↔ (𝑊 ∈ 𝐴 ∧ (𝐹‘𝑊) ≠ ∅)))
59	55, 2, 57, 58	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ∈ (𝐹 supp ∅) ↔ (𝑊 ∈ 𝐴 ∧ (𝐹‘𝑊) ≠ ∅)))
60		simpr 477	. . . . . . . . 9 ⊢ ((𝑊 ∈ 𝐴 ∧ (𝐹‘𝑊) ≠ ∅) → (𝐹‘𝑊) ≠ ∅)
61	59, 60	syl6bi 243	. . . . . . . 8 ⊢ (𝜑 → (𝑊 ∈ (𝐹 supp ∅) → (𝐹‘𝑊) ≠ ∅))
62	41, 61	mpd 15	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑊) ≠ ∅)
63		on0eln0 5780	. . . . . . . 8 ⊢ ((𝐹‘𝑊) ∈ On → (∅ ∈ (𝐹‘𝑊) ↔ (𝐹‘𝑊) ≠ ∅))
64	47, 48, 63	3syl 18	. . . . . . 7 ⊢ (𝜑 → (∅ ∈ (𝐹‘𝑊) ↔ (𝐹‘𝑊) ≠ ∅))
65	62, 64	mpbird 247	. . . . . 6 ⊢ (𝜑 → ∅ ∈ (𝐹‘𝑊))
66	5, 2, 11, 4, 22, 28, 29, 30, 31, 20, 32	cnfcom3lem 8600	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1𝑜))
67		ondif1 7581	. . . . . . . 8 ⊢ (𝑊 ∈ (On ∖ 1𝑜) ↔ (𝑊 ∈ On ∧ ∅ ∈ 𝑊))
68	67	simprbi 480	. . . . . . 7 ⊢ (𝑊 ∈ (On ∖ 1𝑜) → ∅ ∈ 𝑊)
69	66, 68	syl 17	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 𝑊)
70		omabs 7727	. . . . . 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 1327	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑊) ·𝑜 (ω ↑𝑜 𝑊)) = (ω ↑𝑜 𝑊))
72		f1oeq3 6129	. . . . 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 222	. . 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 8599	. . 3 ⊢ (𝜑 → (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹‘𝑊)))
76		f1oco 6159	. . 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 693	. 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 6127	. . 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 224	1 ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→(ω ↑𝑜 𝑊))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729   E cep 5028  ^◡ccnv 5113  dom cdm 5114   ∘ ccom 5118  Oncon0 5723  suc csuc 5725   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065   supp csupp 7295  seq_𝜔cseqom 7542  1_𝑜c1o 7553   +_𝑜 coa 7557   ·_𝑜 comu 7558   ↑_𝑜 coe 7559   finSupp cfsupp 8275  OrdIsocoi 8414   CNF ccnf 8558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-cnf 8559

This theorem is referenced by:  cnfcom3clem  8602