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Theorem cnfcom2 8774
Description: Any nonzero ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-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)
cnfcom2.1 (𝜑 → ∅ ∈ 𝐵)
Assertion
Ref Expression
cnfcom2 (𝜑 → (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)))
Distinct variable groups:   𝑥,𝑘,𝑧,𝐴   𝑥,𝑀   𝑓,𝑘,𝑥,𝑧,𝐹   𝑧,𝑇   𝑥,𝑊   𝑓,𝐺,𝑘,𝑥,𝑧   𝑓,𝐻,𝑥   𝑆,𝑘,𝑧   𝜑,𝑘,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑓)   𝐴(𝑓)   𝐵(𝑥,𝑧,𝑓,𝑘)   𝑆(𝑥,𝑓)   𝑇(𝑥,𝑓,𝑘)   𝐻(𝑧,𝑘)   𝐾(𝑥,𝑧,𝑓,𝑘)   𝑀(𝑧,𝑓,𝑘)   𝑊(𝑧,𝑓,𝑘)

Proof of Theorem cnfcom2
StepHypRef Expression
1 cnfcom.s . . . . 5 𝑆 = dom (ω CNF 𝐴)
2 cnfcom.a . . . . 5 (𝜑𝐴 ∈ On)
3 cnfcom.b . . . . 5 (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))
4 cnfcom.f . . . . 5 𝐹 = ((ω CNF 𝐴)‘𝐵)
5 cnfcom.g . . . . 5 𝐺 = OrdIso( E , (𝐹 supp ∅))
6 cnfcom.h . . . . 5 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)
7 cnfcom.t . . . . 5 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
8 cnfcom.m . . . . 5 𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))
9 cnfcom.k . . . . 5 𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
10 ovex 6842 . . . . . . . . . 10 (𝐹 supp ∅) ∈ V
115oion 8608 . . . . . . . . . 10 ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On)
1210, 11ax-mp 5 . . . . . . . . 9 dom 𝐺 ∈ On
1312elexi 3353 . . . . . . . 8 dom 𝐺 ∈ V
1413uniex 7119 . . . . . . 7 dom 𝐺 ∈ V
1514sucid 5965 . . . . . 6 dom 𝐺 ∈ suc dom 𝐺
16 cnfcom.w . . . . . . 7 𝑊 = (𝐺 dom 𝐺)
17 cnfcom2.1 . . . . . . 7 (𝜑 → ∅ ∈ 𝐵)
181, 2, 3, 4, 5, 6, 7, 8, 9, 16, 17cnfcom2lem 8773 . . . . . 6 (𝜑 → dom 𝐺 = suc dom 𝐺)
1915, 18syl5eleqr 2846 . . . . 5 (𝜑 dom 𝐺 ∈ dom 𝐺)
201, 2, 3, 4, 5, 6, 7, 8, 9, 19cnfcom 8772 . . . 4 (𝜑 → (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 (𝐺 dom 𝐺)) ·𝑜 (𝐹‘(𝐺 dom 𝐺))))
2116oveq2i 6825 . . . . . 6 (ω ↑𝑜 𝑊) = (ω ↑𝑜 (𝐺 dom 𝐺))
2216fveq2i 6356 . . . . . 6 (𝐹𝑊) = (𝐹‘(𝐺 dom 𝐺))
2321, 22oveq12i 6826 . . . . 5 ((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) = ((ω ↑𝑜 (𝐺 dom 𝐺)) ·𝑜 (𝐹‘(𝐺 dom 𝐺)))
24 f1oeq3 6291 . . . . 5 (((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) = ((ω ↑𝑜 (𝐺 dom 𝐺)) ·𝑜 (𝐹‘(𝐺 dom 𝐺))) → ((𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) ↔ (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 (𝐺 dom 𝐺)) ·𝑜 (𝐹‘(𝐺 dom 𝐺)))))
2523, 24ax-mp 5 . . . 4 ((𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) ↔ (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 (𝐺 dom 𝐺)) ·𝑜 (𝐹‘(𝐺 dom 𝐺))))
2620, 25sylibr 224 . . 3 (𝜑 → (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)))
2718fveq2d 6357 . . . 4 (𝜑 → (𝑇‘dom 𝐺) = (𝑇‘suc dom 𝐺))
28 f1oeq1 6289 . . . 4 ((𝑇‘dom 𝐺) = (𝑇‘suc dom 𝐺) → ((𝑇‘dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) ↔ (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))))
2927, 28syl 17 . . 3 (𝜑 → ((𝑇‘dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) ↔ (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))))
3026, 29mpbird 247 . 2 (𝜑 → (𝑇‘dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)))
314fveq2i 6356 . . . . 5 ((ω CNF 𝐴)‘𝐹) = ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵))
32 omelon 8718 . . . . . . 7 ω ∈ On
3332a1i 11 . . . . . 6 (𝜑 → ω ∈ On)
341, 33, 2cantnff1o 8768 . . . . . . . . 9 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴))
35 f1ocnv 6311 . . . . . . . . 9 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴) → (ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆)
36 f1of 6299 . . . . . . . . 9 ((ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
3734, 35, 363syl 18 . . . . . . . 8 (𝜑(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
3837, 3ffvelrnd 6524 . . . . . . 7 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
394, 38syl5eqel 2843 . . . . . 6 (𝜑𝐹𝑆)
408oveq1i 6824 . . . . . . . . . 10 (𝑀 +𝑜 𝑧) = (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)
4140a1i 11 . . . . . . . . 9 ((𝑘 ∈ V ∧ 𝑧 ∈ V) → (𝑀 +𝑜 𝑧) = (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
4241mpt2eq3ia 6886 . . . . . . . 8 (𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧))
43 eqid 2760 . . . . . . . 8 ∅ = ∅
44 seqomeq12 7719 . . . . . . . 8 (((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)) ∧ ∅ = ∅) → seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅))
4542, 43, 44mp2an 710 . . . . . . 7 seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
466, 45eqtri 2782 . . . . . 6 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘))) +𝑜 𝑧)), ∅)
471, 33, 2, 5, 39, 46cantnfval 8740 . . . . 5 (𝜑 → ((ω CNF 𝐴)‘𝐹) = (𝐻‘dom 𝐺))
4831, 47syl5reqr 2809 . . . 4 (𝜑 → (𝐻‘dom 𝐺) = ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵)))
4918fveq2d 6357 . . . 4 (𝜑 → (𝐻‘dom 𝐺) = (𝐻‘suc dom 𝐺))
50 f1ocnvfv2 6697 . . . . 5 (((ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴) ∧ 𝐵 ∈ (ω ↑𝑜 𝐴)) → ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵)) = 𝐵)
5134, 3, 50syl2anc 696 . . . 4 (𝜑 → ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵)) = 𝐵)
5248, 49, 513eqtr3d 2802 . . 3 (𝜑 → (𝐻‘suc dom 𝐺) = 𝐵)
53 f1oeq2 6290 . . 3 ((𝐻‘suc dom 𝐺) = 𝐵 → ((𝑇‘dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) ↔ (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))))
5452, 53syl 17 . 2 (𝜑 → ((𝑇‘dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)) ↔ (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊))))
5530, 54mpbid 222 1 (𝜑 → (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (𝐹𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cun 3713  c0 4058   cuni 4588  cmpt 4881   E cep 5178  ccnv 5265  dom cdm 5266  Oncon0 5884  suc csuc 5886  wf 6045  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6814  cmpt2 6816  ωcom 7231   supp csupp 7464  seq𝜔cseqom 7712   +𝑜 coa 7727   ·𝑜 comu 7728  𝑜 coe 7729  OrdIsocoi 8581   CNF ccnf 8733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-supp 7465  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-seqom 7713  df-1o 7730  df-2o 7731  df-oadd 7734  df-omul 7735  df-oexp 7736  df-er 7913  df-map 8027  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-fsupp 8443  df-oi 8582  df-cnf 8734
This theorem is referenced by:  cnfcom3  8776
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