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Theorem cnfcom3 8765
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 8767.) (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 8707 . . . . . 6 ω ∈ On
2 cnfcom.a . . . . . . 7 (𝜑𝐴 ∈ On)
3 suppssdm 7459 . . . . . . . . 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 8757 . . . . . . . . . . . . . . 15 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴))
8 f1ocnv 6290 . . . . . . . . . . . . . . 15 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴) → (ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆)
9 f1of 6278 . . . . . . . . . . . . . . 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 6503 . . . . . . . . . . . . 13 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
134, 12syl5eqel 2854 . . . . . . . . . . . 12 (𝜑𝐹𝑆)
145, 6, 2cantnfs 8727 . . . . . . . . . . . 12 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
1513, 14mpbid 222 . . . . . . . . . . 11 (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
1615simpld 482 . . . . . . . . . 10 (𝜑𝐹:𝐴⟶ω)
17 fdm 6191 . . . . . . . . . 10 (𝐹:𝐴⟶ω → dom 𝐹 = 𝐴)
1816, 17syl 17 . . . . . . . . 9 (𝜑 → dom 𝐹 = 𝐴)
193, 18syl5sseq 3802 . . . . . . . 8 (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
20 cnfcom.w . . . . . . . . 9 𝑊 = (𝐺 dom 𝐺)
21 ovex 6823 . . . . . . . . . . . . . . 15 (𝐹 supp ∅) ∈ V
22 cnfcom.g . . . . . . . . . . . . . . . 16 𝐺 = OrdIso( E , (𝐹 supp ∅))
2322oion 8597 . . . . . . . . . . . . . . 15 ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On)
2421, 23ax-mp 5 . . . . . . . . . . . . . 14 dom 𝐺 ∈ On
2524elexi 3365 . . . . . . . . . . . . 13 dom 𝐺 ∈ V
2625uniex 7100 . . . . . . . . . . . 12 dom 𝐺 ∈ V
2726sucid 5947 . . . . . . . . . . 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 7232 . . . . . . . . . . . . . 14 ∅ ∈ ω
3433a1i 11 . . . . . . . . . . . . 13 (𝜑 → ∅ ∈ ω)
3532, 34sseldd 3753 . . . . . . . . . . . 12 (𝜑 → ∅ ∈ 𝐵)
365, 2, 11, 4, 22, 28, 29, 30, 31, 20, 35cnfcom2lem 8762 . . . . . . . . . . 11 (𝜑 → dom 𝐺 = suc dom 𝐺)
3727, 36syl5eleqr 2857 . . . . . . . . . 10 (𝜑 dom 𝐺 ∈ dom 𝐺)
3822oif 8591 . . . . . . . . . . 11 𝐺:dom 𝐺⟶(𝐹 supp ∅)
3938ffvelrni 6501 . . . . . . . . . 10 ( dom 𝐺 ∈ dom 𝐺 → (𝐺 dom 𝐺) ∈ (𝐹 supp ∅))
4037, 39syl 17 . . . . . . . . 9 (𝜑 → (𝐺 dom 𝐺) ∈ (𝐹 supp ∅))
4120, 40syl5eqel 2854 . . . . . . . 8 (𝜑𝑊 ∈ (𝐹 supp ∅))
4219, 41sseldd 3753 . . . . . . 7 (𝜑𝑊𝐴)
43 onelon 5891 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑊𝐴) → 𝑊 ∈ On)
442, 42, 43syl2anc 573 . . . . . 6 (𝜑𝑊 ∈ On)
45 oecl 7771 . . . . . 6 ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑𝑜 𝑊) ∈ On)
461, 44, 45sylancr 575 . . . . 5 (𝜑 → (ω ↑𝑜 𝑊) ∈ On)
4716, 42ffvelrnd 6503 . . . . . 6 (𝜑 → (𝐹𝑊) ∈ ω)
48 nnon 7218 . . . . . 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 8219 . . . . 5 (((ω ↑𝑜 𝑊) ∈ On ∧ (𝐹𝑊) ∈ On) → (𝑋𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))–1-1-onto→((𝐹𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
5346, 49, 52syl2anc 573 . . . 4 (𝜑 → (𝑋𝑌):((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))–1-1-onto→((𝐹𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
54 ffn 6185 . . . . . . . . . . 11 (𝐹:𝐴⟶ω → 𝐹 Fn 𝐴)
5516, 54syl 17 . . . . . . . . . 10 (𝜑𝐹 Fn 𝐴)
56 0ex 4924 . . . . . . . . . . 11 ∅ ∈ V
5756a1i 11 . . . . . . . . . 10 (𝜑 → ∅ ∈ V)
58 elsuppfn 7454 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝐴 ∈ On ∧ ∅ ∈ V) → (𝑊 ∈ (𝐹 supp ∅) ↔ (𝑊𝐴 ∧ (𝐹𝑊) ≠ ∅)))
5955, 2, 57, 58syl3anc 1476 . . . . . . . . 9 (𝜑 → (𝑊 ∈ (𝐹 supp ∅) ↔ (𝑊𝐴 ∧ (𝐹𝑊) ≠ ∅)))
60 simpr 471 . . . . . . . . 9 ((𝑊𝐴 ∧ (𝐹𝑊) ≠ ∅) → (𝐹𝑊) ≠ ∅)
6159, 60syl6bi 243 . . . . . . . 8 (𝜑 → (𝑊 ∈ (𝐹 supp ∅) → (𝐹𝑊) ≠ ∅))
6241, 61mpd 15 . . . . . . 7 (𝜑 → (𝐹𝑊) ≠ ∅)
63 on0eln0 5923 . . . . . . . 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 8764 . . . . . . 7 (𝜑𝑊 ∈ (On ∖ 1𝑜))
67 ondif1 7735 . . . . . . . 8 (𝑊 ∈ (On ∖ 1𝑜) ↔ (𝑊 ∈ On ∧ ∅ ∈ 𝑊))
6867simprbi 484 . . . . . . 7 (𝑊 ∈ (On ∖ 1𝑜) → ∅ ∈ 𝑊)
6966, 68syl 17 . . . . . 6 (𝜑 → ∅ ∈ 𝑊)
70 omabs 7881 . . . . . 6 ((((𝐹𝑊) ∈ ω ∧ ∅ ∈ (𝐹𝑊)) ∧ (𝑊 ∈ On ∧ ∅ ∈ 𝑊)) → ((𝐹𝑊) ·𝑜 (ω ↑𝑜 𝑊)) = (ω ↑𝑜 𝑊))
7147, 65, 44, 69, 70syl22anc 1477 . . . . 5 (𝜑 → ((𝐹𝑊) ·𝑜 (ω ↑𝑜 𝑊)) = (ω ↑𝑜 𝑊))
72 f1oeq3 6270 . . . . 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 8763 . . 3 (𝜑 → (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)))
76 f1oco 6300 . . 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 573 . 2 (𝜑 → ((𝑋𝑌) ∘ (𝑇‘dom 𝐺)):𝐵1-1-onto→(ω ↑𝑜 𝑊))
78 cnfcom.n . . 3 𝑁 = ((𝑋𝑌) ∘ (𝑇‘dom 𝐺))
79 f1oeq1 6268 . . 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 382   = wceq 1631  wcel 2145  wne 2943  Vcvv 3351  cdif 3720  cun 3721  wss 3723  c0 4063   cuni 4574   class class class wbr 4786  cmpt 4863   E cep 5161  ccnv 5248  dom cdm 5249  ccom 5253  Oncon0 5866  suc csuc 5868   Fn wfn 6026  wf 6027  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  cmpt2 6795  ωcom 7212   supp csupp 7446  seq𝜔cseqom 7695  1𝑜c1o 7706   +𝑜 coa 7710   ·𝑜 comu 7711  𝑜 coe 7712   finSupp cfsupp 8431  OrdIsocoi 8570   CNF ccnf 8722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-supp 7447  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-seqom 7696  df-1o 7713  df-2o 7714  df-oadd 7717  df-omul 7718  df-oexp 7719  df-er 7896  df-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-fsupp 8432  df-oi 8571  df-cnf 8723
This theorem is referenced by:  cnfcom3clem  8766
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