Theorem cnfcom3 8898

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 8900.) (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 8840	. . . . . 6 ⊢ ω ∈ On
2		cnfcom.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
3		suppssdm 7589	. . . . . . . . 9 ⊢ (𝐹 supp ∅) ⊆ dom 𝐹
4		cnfcom.f	. . . . . . . . . . . 12 ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)
5		cnfcom.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = dom (ω CNF 𝐴)
6	1	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ω ∈ On)
7	5, 6, 2	cantnff1o 8890	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ω CNF 𝐴):𝑆–1-1-onto→(ω ↑o 𝐴))
8		f1ocnv 6403	. . . . . . . . . . . . . 14 ⊢ ((ω CNF 𝐴):𝑆–1-1-onto→(ω ↑o 𝐴) → ◡(ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto→𝑆)
9		f1of 6391	. . . . . . . . . . . . . 14 ⊢ (◡(ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto→𝑆 → ◡(ω CNF 𝐴):(ω ↑o 𝐴)⟶𝑆)
10	7, 8, 9	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ◡(ω CNF 𝐴):(ω ↑o 𝐴)⟶𝑆)
11		cnfcom.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))
12	10, 11	ffvelrnd 6624	. . . . . . . . . . . 12 ⊢ (𝜑 → (◡(ω CNF 𝐴)‘𝐵) ∈ 𝑆)
13	4, 12	syl5eqel 2863	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
14	5, 6, 2	cantnfs 8860	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
15	13, 14	mpbid 224	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
16	15	simpld 490	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶ω)
17	3, 16	fssdm 6307	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
18		cnfcom.w	. . . . . . . . 9 ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)
19		ovex 6954	. . . . . . . . . . . . . . 15 ⊢ (𝐹 supp ∅) ∈ V
20		cnfcom.g	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
21	20	oion 8730	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On)
22	19, 21	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ dom 𝐺 ∈ On
23	22	elexi 3415	. . . . . . . . . . . . 13 ⊢ dom 𝐺 ∈ V
24	23	uniex 7230	. . . . . . . . . . . 12 ⊢ ∪ dom 𝐺 ∈ V
25	24	sucid 6055	. . . . . . . . . . 11 ⊢ ∪ dom 𝐺 ∈ suc ∪ dom 𝐺
26		cnfcom.h	. . . . . . . . . . . 12 ⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)
27		cnfcom.t	. . . . . . . . . . . 12 ⊢ 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
28		cnfcom.m	. . . . . . . . . . . 12 ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))
29		cnfcom.k	. . . . . . . . . . . 12 ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))
30		cnfcom3.1	. . . . . . . . . . . . 13 ⊢ (𝜑 → ω ⊆ 𝐵)
31		peano1 7363	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ ω
32	31	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∅ ∈ ω)
33	30, 32	sseldd 3822	. . . . . . . . . . . 12 ⊢ (𝜑 → ∅ ∈ 𝐵)
34	5, 2, 11, 4, 20, 26, 27, 28, 29, 18, 33	cnfcom2lem 8895	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐺 = suc ∪ dom 𝐺)
35	25, 34	syl5eleqr 2866	. . . . . . . . . 10 ⊢ (𝜑 → ∪ dom 𝐺 ∈ dom 𝐺)
36	20	oif 8724	. . . . . . . . . . 11 ⊢ 𝐺:dom 𝐺⟶(𝐹 supp ∅)
37	36	ffvelrni 6622	. . . . . . . . . 10 ⊢ (∪ dom 𝐺 ∈ dom 𝐺 → (𝐺‘∪ dom 𝐺) ∈ (𝐹 supp ∅))
38	35, 37	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘∪ dom 𝐺) ∈ (𝐹 supp ∅))
39	18, 38	syl5eqel 2863	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ (𝐹 supp ∅))
40	17, 39	sseldd 3822	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐴)
41		onelon 6001	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑊 ∈ 𝐴) → 𝑊 ∈ On)
42	2, 40, 41	syl2anc 579	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ On)
43		oecl 7901	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑o 𝑊) ∈ On)
44	1, 42, 43	sylancr 581	. . . . 5 ⊢ (𝜑 → (ω ↑o 𝑊) ∈ On)
45	16, 40	ffvelrnd 6624	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑊) ∈ ω)
46		nnon 7349	. . . . . 6 ⊢ ((𝐹‘𝑊) ∈ ω → (𝐹‘𝑊) ∈ On)
47	45, 46	syl 17	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑊) ∈ On)
48		cnfcom.y	. . . . . 6 ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑢) +o 𝑣))
49		cnfcom.x	. . . . . 6 ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((𝐹‘𝑊) ·o 𝑣) +o 𝑢))
50	48, 49	omf1o 8351	. . . . 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)))
51	44, 47, 50	syl2anc 579	. . . 4 ⊢ (𝜑 → (𝑋 ∘ ◡𝑌):((ω ↑o 𝑊) ·o (𝐹‘𝑊))–1-1-onto→((𝐹‘𝑊) ·o (ω ↑o 𝑊)))
52	16	ffnd 6292	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 Fn 𝐴)
53		0ex 5026	. . . . . . . . . . 11 ⊢ ∅ ∈ V
54	53	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ V)
55		elsuppfn 7584	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ On ∧ ∅ ∈ V) → (𝑊 ∈ (𝐹 supp ∅) ↔ (𝑊 ∈ 𝐴 ∧ (𝐹‘𝑊) ≠ ∅)))
56	52, 2, 54, 55	syl3anc 1439	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ∈ (𝐹 supp ∅) ↔ (𝑊 ∈ 𝐴 ∧ (𝐹‘𝑊) ≠ ∅)))
57		simpr 479	. . . . . . . . 9 ⊢ ((𝑊 ∈ 𝐴 ∧ (𝐹‘𝑊) ≠ ∅) → (𝐹‘𝑊) ≠ ∅)
58	56, 57	syl6bi 245	. . . . . . . 8 ⊢ (𝜑 → (𝑊 ∈ (𝐹 supp ∅) → (𝐹‘𝑊) ≠ ∅))
59	39, 58	mpd 15	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝑊) ≠ ∅)
60		on0eln0 6031	. . . . . . . 8 ⊢ ((𝐹‘𝑊) ∈ On → (∅ ∈ (𝐹‘𝑊) ↔ (𝐹‘𝑊) ≠ ∅))
61	45, 46, 60	3syl 18	. . . . . . 7 ⊢ (𝜑 → (∅ ∈ (𝐹‘𝑊) ↔ (𝐹‘𝑊) ≠ ∅))
62	59, 61	mpbird 249	. . . . . 6 ⊢ (𝜑 → ∅ ∈ (𝐹‘𝑊))
63	5, 2, 11, 4, 20, 26, 27, 28, 29, 18, 30	cnfcom3lem 8897	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1o))
64		ondif1 7865	. . . . . . . 8 ⊢ (𝑊 ∈ (On ∖ 1o) ↔ (𝑊 ∈ On ∧ ∅ ∈ 𝑊))
65	64	simprbi 492	. . . . . . 7 ⊢ (𝑊 ∈ (On ∖ 1o) → ∅ ∈ 𝑊)
66	63, 65	syl 17	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 𝑊)
67		omabs 8011	. . . . . 6 ⊢ ((((𝐹‘𝑊) ∈ ω ∧ ∅ ∈ (𝐹‘𝑊)) ∧ (𝑊 ∈ On ∧ ∅ ∈ 𝑊)) → ((𝐹‘𝑊) ·o (ω ↑o 𝑊)) = (ω ↑o 𝑊))
68	45, 62, 42, 66, 67	syl22anc 829	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑊) ·o (ω ↑o 𝑊)) = (ω ↑o 𝑊))
69	68	f1oeq3d 6388	. . . 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)))
70	51, 69	mpbid 224	. . 3 ⊢ (𝜑 → (𝑋 ∘ ◡𝑌):((ω ↑o 𝑊) ·o (𝐹‘𝑊))–1-1-onto→(ω ↑o 𝑊))
71	5, 2, 11, 4, 20, 26, 27, 28, 29, 18, 33	cnfcom2 8896	. . 3 ⊢ (𝜑 → (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊)))
72		f1oco 6413	. . 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))
73	70, 71, 72	syl2anc 579	. 2 ⊢ (𝜑 → ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺)):𝐵–1-1-onto→(ω ↑o 𝑊))
74		cnfcom.n	. . 3 ⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺))
75		f1oeq1 6380	. . 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)))
76	74, 75	ax-mp 5	. 2 ⊢ (𝑁:𝐵–1-1-onto→(ω ↑o 𝑊) ↔ ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺)):𝐵–1-1-onto→(ω ↑o 𝑊))
77	73, 76	sylibr 226	1 ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→(ω ↑o 𝑊))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  Vcvv 3398   ∖ cdif 3789   ∪ cun 3790   ⊆ wss 3792  ∅c0 4141  ∪ cuni 4671   class class class wbr 4886   ↦ cmpt 4965   E cep 5265  ^◡ccnv 5354  dom cdm 5355   ∘ ccom 5359  Oncon0 5976  suc csuc 5978   Fn wfn 6130  ⟶wf 6131  –1-1-onto→wf1o 6134  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  ωcom 7343   supp csupp 7576  seq_𝜔cseqom 7825  1_oc1o 7836   +_o coa 7840   ·_o comu 7841   ↑_o coe 7842   finSupp cfsupp 8563  OrdIsocoi 8703   CNF ccnf 8855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-seqom 7826  df-1o 7843  df-2o 7844  df-oadd 7847  df-omul 7848  df-oexp 7849  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-oi 8704  df-cnf 8856

This theorem is referenced by:  cnfcom3clem  8899