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Theorem cnfcom2 9149
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 7168 . . . . . . . . . 10 (𝐹 supp ∅) ∈ V
115oion 8984 . . . . . . . . . 10 ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On)
1210, 11ax-mp 5 . . . . . . . . 9 dom 𝐺 ∈ On
1312elexi 3460 . . . . . . . 8 dom 𝐺 ∈ V
1413uniex 7447 . . . . . . 7 dom 𝐺 ∈ V
1514sucid 6238 . . . . . 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 9148 . . . . . 6 (𝜑 → dom 𝐺 = suc dom 𝐺)
1915, 18eleqtrrid 2897 . . . . 5 (𝜑 dom 𝐺 ∈ dom 𝐺)
201, 2, 3, 4, 5, 6, 7, 8, 9, 19cnfcom 9147 . . . 4 (𝜑 → (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑o (𝐺 dom 𝐺)) ·o (𝐹‘(𝐺 dom 𝐺))))
2116oveq2i 7146 . . . . . 6 (ω ↑o 𝑊) = (ω ↑o (𝐺 dom 𝐺))
2216fveq2i 6648 . . . . . 6 (𝐹𝑊) = (𝐹‘(𝐺 dom 𝐺))
2321, 22oveq12i 7147 . . . . 5 ((ω ↑o 𝑊) ·o (𝐹𝑊)) = ((ω ↑o (𝐺 dom 𝐺)) ·o (𝐹‘(𝐺 dom 𝐺)))
24 f1oeq3 6581 . . . . 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 237 . . 3 (𝜑 → (𝑇‘suc dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑o 𝑊) ·o (𝐹𝑊)))
2718fveq2d 6649 . . . 4 (𝜑 → (𝑇‘dom 𝐺) = (𝑇‘suc dom 𝐺))
28 f1oeq1 6579 . . . 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 260 . 2 (𝜑 → (𝑇‘dom 𝐺):(𝐻‘suc dom 𝐺)–1-1-onto→((ω ↑o 𝑊) ·o (𝐹𝑊)))
314fveq2i 6648 . . . . 5 ((ω CNF 𝐴)‘𝐹) = ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵))
32 omelon 9093 . . . . . . 7 ω ∈ On
3332a1i 11 . . . . . 6 (𝜑 → ω ∈ On)
341, 33, 2cantnff1o 9143 . . . . . . . . 9 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑o 𝐴))
35 f1ocnv 6602 . . . . . . . . 9 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑o 𝐴) → (ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto𝑆)
36 f1of 6590 . . . . . . . . 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 6829 . . . . . . 7 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
394, 38eqeltrid 2894 . . . . . 6 (𝜑𝐹𝑆)
408oveq1i 7145 . . . . . . . . . 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))
4241mpoeq3ia 7211 . . . . . . . 8 (𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)) = (𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧))
43 eqid 2798 . . . . . . . 8 ∅ = ∅
44 seqomeq12 8073 . . . . . . . 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 691 . . . . . . 7 seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)
466, 45eqtri 2821 . . . . . 6 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (𝐺𝑘)) ·o (𝐹‘(𝐺𝑘))) +o 𝑧)), ∅)
471, 33, 2, 5, 39, 46cantnfval 9115 . . . . 5 (𝜑 → ((ω CNF 𝐴)‘𝐹) = (𝐻‘dom 𝐺))
4831, 47syl5reqr 2848 . . . 4 (𝜑 → (𝐻‘dom 𝐺) = ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵)))
4918fveq2d 6649 . . . 4 (𝜑 → (𝐻‘dom 𝐺) = (𝐻‘suc dom 𝐺))
50 f1ocnvfv2 7012 . . . . 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 587 . . . 4 (𝜑 → ((ω CNF 𝐴)‘((ω CNF 𝐴)‘𝐵)) = 𝐵)
5248, 49, 513eqtr3d 2841 . . 3 (𝜑 → (𝐻‘suc dom 𝐺) = 𝐵)
5352f1oeq2d 6586 . 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))))
5430, 53mpbid 235 1 (𝜑 → (𝑇‘dom 𝐺):𝐵1-1-onto→((ω ↑o 𝑊) ·o (𝐹𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  Vcvv 3441  cun 3879  c0 4243   cuni 4800  cmpt 5110   E cep 5429  ccnv 5518  dom cdm 5519  Oncon0 6159  suc csuc 6161  wf 6320  1-1-ontowf1o 6323  cfv 6324  (class class class)co 7135  cmpo 7137  ωcom 7560   supp csupp 7813  seqωcseqom 8066   +o coa 8082   ·o comu 8083  o coe 8084  OrdIsocoi 8957   CNF ccnf 9108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-seqom 8067  df-1o 8085  df-2o 8086  df-oadd 8089  df-omul 8090  df-oexp 8091  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-oi 8958  df-cnf 9109
This theorem is referenced by:  cnfcom3  9151
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