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Theorem cnfcom3 8586
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 8588.) (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 (𝜑𝐴 ∈ On)
cnfcom.b (𝜑𝐵 ∈ (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 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))
cnfcom.k 𝐾 = ((𝑥𝑀 ↦ (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 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((𝐹𝑊) ·𝑜 𝑣) +𝑜 𝑢))
cnfcom.y 𝑌 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑢) +𝑜 𝑣))
cnfcom.n 𝑁 = ((𝑋𝑌) ∘ (𝑇‘dom 𝐺))
Assertion
Ref Expression
cnfcom3 (𝜑𝑁:𝐵1-1-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
StepHypRef Expression
1 omelon 8528 . . . . . 6 ω ∈ On
2 cnfcom.a . . . . . . 7 (𝜑𝐴 ∈ On)
3 suppssdm 7293 . . . . . . . . 9 (𝐹 supp ∅) ⊆ dom 𝐹
4 cnfcom.f . . . . . . . . . . . . 13 𝐹 = ((ω CNF 𝐴)‘𝐵)
5 cnfcom.s . . . . . . . . . . . . . . . 16 𝑆 = dom (ω CNF 𝐴)
61a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → ω ∈ On)
75, 6, 2cantnff1o 8578 . . . . . . . . . . . . . . 15 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴))
8 f1ocnv 6136 . . . . . . . . . . . . . . 15 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴) → (ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆)
9 f1of 6124 . . . . . . . . . . . . . . 15 ((ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
107, 8, 93syl 18 . . . . . . . . . . . . . 14 (𝜑(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
11 cnfcom.b . . . . . . . . . . . . . 14 (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))
1210, 11ffvelrnd 6346 . . . . . . . . . . . . 13 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
134, 12syl5eqel 2703 . . . . . . . . . . . 12 (𝜑𝐹𝑆)
145, 6, 2cantnfs 8548 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
1513, 14mpbid 222 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
1615simpld 475 . . . . . . . . . 10 (𝜑𝐹:𝐴⟶ω)
17 fdm 6038 . . . . . . . . . 10 (𝐹:𝐴⟶ω → dom 𝐹 = 𝐴)
1816, 17syl 17 . . . . . . . . 9 (𝜑 → dom 𝐹 = 𝐴)
193, 18syl5sseq 3645 . . . . . . . 8 (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
20 cnfcom.w . . . . . . . . 9 𝑊 = (𝐺 dom 𝐺)
21 ovex 6663 . . . . . . . . . . . . . . 15 (𝐹 supp ∅) ∈ V
22 cnfcom.g . . . . . . . . . . . . . . . 16 𝐺 = OrdIso( E , (𝐹 supp ∅))
2322oion 8426 . . . . . . . . . . . . . . 15 ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On)
2421, 23ax-mp 5 . . . . . . . . . . . . . 14 dom 𝐺 ∈ On
2524elexi 3208 . . . . . . . . . . . . 13 dom 𝐺 ∈ V
2625uniex 6938 . . . . . . . . . . . 12 dom 𝐺 ∈ V
2726sucid 5792 . . . . . . . . . . 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 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))
31 cnfcom.k . . . . . . . . . . . 12 𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
32 cnfcom3.1 . . . . . . . . . . . . 13 (𝜑 → ω ⊆ 𝐵)
33 peano1 7070 . . . . . . . . . . . . . 14 ∅ ∈ ω
3433a1i 11 . . . . . . . . . . . . 13 (𝜑 → ∅ ∈ ω)
3532, 34sseldd 3596 . . . . . . . . . . . 12 (𝜑 → ∅ ∈ 𝐵)
365, 2, 11, 4, 22, 28, 29, 30, 31, 20, 35cnfcom2lem 8583 . . . . . . . . . . 11 (𝜑 → dom 𝐺 = suc dom 𝐺)
3727, 36syl5eleqr 2706 . . . . . . . . . 10 (𝜑 dom 𝐺 ∈ dom 𝐺)
3822oif 8420 . . . . . . . . . . 11 𝐺:dom 𝐺⟶(𝐹 supp ∅)
3938ffvelrni 6344 . . . . . . . . . 10 ( dom 𝐺 ∈ dom 𝐺 → (𝐺 dom 𝐺) ∈ (𝐹 supp ∅))
4037, 39syl 17 . . . . . . . . 9 (𝜑 → (𝐺 dom 𝐺) ∈ (𝐹 supp ∅))
4120, 40syl5eqel 2703 . . . . . . . 8 (𝜑𝑊 ∈ (𝐹 supp ∅))
4219, 41sseldd 3596 . . . . . . 7 (𝜑𝑊𝐴)
43 onelon 5736 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑊𝐴) → 𝑊 ∈ On)
442, 42, 43syl2anc 692 . . . . . 6 (𝜑𝑊 ∈ On)
45 oecl 7602 . . . . . 6 ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑𝑜 𝑊) ∈ On)
461, 44, 45sylancr 694 . . . . 5 (𝜑 → (ω ↑𝑜 𝑊) ∈ On)
4716, 42ffvelrnd 6346 . . . . . 6 (𝜑 → (𝐹𝑊) ∈ ω)
48 nnon 7056 . . . . . 6 ((𝐹𝑊) ∈ ω → (𝐹𝑊) ∈ On)
4947, 48syl 17 . . . . 5 (𝜑 → (𝐹𝑊) ∈ On)
50 cnfcom.y . . . . . 6 𝑌 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((ω ↑𝑜 𝑊) ·𝑜 𝑢) +𝑜 𝑣))
51 cnfcom.x . . . . . 6 𝑋 = (𝑢 ∈ (𝐹𝑊), 𝑣 ∈ (ω ↑𝑜 𝑊) ↦ (((𝐹𝑊) ·𝑜 𝑣) +𝑜 𝑢))
5250, 51omf1o 8048 . . . . 5 (((ω ↑𝑜 𝑊) ∈ On ∧ (𝐹𝑊) ∈ On) → (𝑋𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))–1-1-onto→((𝐹𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
5346, 49, 52syl2anc 692 . . . 4 (𝜑 → (𝑋𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))–1-1-onto→((𝐹𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
54 ffn 6032 . . . . . . . . . . 11 (𝐹:𝐴⟶ω → 𝐹 Fn 𝐴)
5516, 54syl 17 . . . . . . . . . 10 (𝜑𝐹 Fn 𝐴)
56 0ex 4781 . . . . . . . . . . 11 ∅ ∈ V
5756a1i 11 . . . . . . . . . 10 (𝜑 → ∅ ∈ V)
58 elsuppfn 7288 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝐴 ∈ On ∧ ∅ ∈ V) → (𝑊 ∈ (𝐹 supp ∅) ↔ (𝑊𝐴 ∧ (𝐹𝑊) ≠ ∅)))
5955, 2, 57, 58syl3anc 1324 . . . . . . . . 9 (𝜑 → (𝑊 ∈ (𝐹 supp ∅) ↔ (𝑊𝐴 ∧ (𝐹𝑊) ≠ ∅)))
60 simpr 477 . . . . . . . . 9 ((𝑊𝐴 ∧ (𝐹𝑊) ≠ ∅) → (𝐹𝑊) ≠ ∅)
6159, 60syl6bi 243 . . . . . . . 8 (𝜑 → (𝑊 ∈ (𝐹 supp ∅) → (𝐹𝑊) ≠ ∅))
6241, 61mpd 15 . . . . . . 7 (𝜑 → (𝐹𝑊) ≠ ∅)
63 on0eln0 5768 . . . . . . . 8 ((𝐹𝑊) ∈ On → (∅ ∈ (𝐹𝑊) ↔ (𝐹𝑊) ≠ ∅))
6447, 48, 633syl 18 . . . . . . 7 (𝜑 → (∅ ∈ (𝐹𝑊) ↔ (𝐹𝑊) ≠ ∅))
6562, 64mpbird 247 . . . . . 6 (𝜑 → ∅ ∈ (𝐹𝑊))
665, 2, 11, 4, 22, 28, 29, 30, 31, 20, 32cnfcom3lem 8585 . . . . . . 7 (𝜑𝑊 ∈ (On ∖ 1𝑜))
67 ondif1 7566 . . . . . . . 8 (𝑊 ∈ (On ∖ 1𝑜) ↔ (𝑊 ∈ On ∧ ∅ ∈ 𝑊))
6867simprbi 480 . . . . . . 7 (𝑊 ∈ (On ∖ 1𝑜) → ∅ ∈ 𝑊)
6966, 68syl 17 . . . . . 6 (𝜑 → ∅ ∈ 𝑊)
70 omabs 7712 . . . . . 6 ((((𝐹𝑊) ∈ ω ∧ ∅ ∈ (𝐹𝑊)) ∧ (𝑊 ∈ On ∧ ∅ ∈ 𝑊)) → ((𝐹𝑊) ·𝑜 (ω ↑𝑜 𝑊)) = (ω ↑𝑜 𝑊))
7147, 65, 44, 69, 70syl22anc 1325 . . . . 5 (𝜑 → ((𝐹𝑊) ·𝑜 (ω ↑𝑜 𝑊)) = (ω ↑𝑜 𝑊))
72 f1oeq3 6116 . . . . 5 (((𝐹𝑊) ·𝑜 (ω ↑𝑜 𝑊)) = (ω ↑𝑜 𝑊) → ((𝑋𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))–1-1-onto→((𝐹𝑊) ·𝑜 (ω ↑𝑜 𝑊)) ↔ (𝑋𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))–1-1-onto→(ω ↑𝑜 𝑊)))
7371, 72syl 17 . . . 4 (𝜑 → ((𝑋𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))–1-1-onto→((𝐹𝑊) ·𝑜 (ω ↑𝑜 𝑊)) ↔ (𝑋𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))–1-1-onto→(ω ↑𝑜 𝑊)))
7453, 73mpbid 222 . . 3 (𝜑 → (𝑋𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))–1-1-onto→(ω ↑𝑜 𝑊))
755, 2, 11, 4, 22, 28, 29, 30, 31, 20, 35cnfcom2 8584 . . 3 (𝜑 → (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)))
76 f1oco 6146 . . 3 (((𝑋𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))–1-1-onto→(ω ↑𝑜 𝑊) ∧ (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))) → ((𝑋𝑌) ∘ (𝑇‘dom 𝐺)):𝐵1-1-onto→(ω ↑𝑜 𝑊))
7774, 75, 76syl2anc 692 . 2 (𝜑 → ((𝑋𝑌) ∘ (𝑇‘dom 𝐺)):𝐵1-1-onto→(ω ↑𝑜 𝑊))
78 cnfcom.n . . 3 𝑁 = ((𝑋𝑌) ∘ (𝑇‘dom 𝐺))
79 f1oeq1 6114 . . 3 (𝑁 = ((𝑋𝑌) ∘ (𝑇‘dom 𝐺)) → (𝑁:𝐵1-1-onto→(ω ↑𝑜 𝑊) ↔ ((𝑋𝑌) ∘ (𝑇‘dom 𝐺)):𝐵1-1-onto→(ω ↑𝑜 𝑊)))
8078, 79ax-mp 5 . 2 (𝑁:𝐵1-1-onto→(ω ↑𝑜 𝑊) ↔ ((𝑋𝑌) ∘ (𝑇‘dom 𝐺)):𝐵1-1-onto→(ω ↑𝑜 𝑊))
8177, 80sylibr 224 1 (𝜑𝑁:𝐵1-1-onto→(ω ↑𝑜 𝑊))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wne 2791  Vcvv 3195  cdif 3564  cun 3565  wss 3567  c0 3907   cuni 4427   class class class wbr 4644  cmpt 4720   E cep 5018  ccnv 5103  dom cdm 5104  ccom 5108  Oncon0 5711  suc csuc 5713   Fn wfn 5871  wf 5872  1-1-ontowf1o 5875  cfv 5876  (class class class)co 6635  cmpt2 6637  ωcom 7050   supp csupp 7280  seq𝜔cseqom 7527  1𝑜c1o 7538   +𝑜 coa 7542   ·𝑜 comu 7543  𝑜 coe 7544   finSupp cfsupp 8260  OrdIsocoi 8399   CNF ccnf 8543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-supp 7281  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-seqom 7528  df-1o 7545  df-2o 7546  df-oadd 7549  df-omul 7550  df-oexp 7551  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fsupp 8261  df-oi 8400  df-cnf 8544
This theorem is referenced by:  cnfcom3clem  8587
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