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Theorem cnfcom3c 9746
Description: Wrap the construction of cnfcom3 9744 into an existential quantifier. For any ω ⊆ 𝑏, there is a bijection from 𝑏 to some power of ω. Furthermore, this bijection is canonical , which means that we can find a single function 𝑔 which will give such bijections for every 𝑏 less than some arbitrarily large bound 𝐴. (Contributed by Mario Carneiro, 30-May-2015.)
Assertion
Ref Expression
cnfcom3c (𝐴 ∈ On → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
Distinct variable group:   𝑔,𝑏,𝑤,𝐴

Proof of Theorem cnfcom3c
Dummy variables 𝑓 𝑘 𝑢 𝑣 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . 2 dom (ω CNF 𝐴) = dom (ω CNF 𝐴)
2 eqid 2737 . 2 ((ω CNF 𝐴)‘𝑏) = ((ω CNF 𝐴)‘𝑏)
3 eqid 2737 . 2 OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)) = OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))
4 eqid 2737 . 2 seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) +o 𝑧)), ∅)
5 eqid 2737 . 2 seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ ((𝑥 ∈ ((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) ↦ (dom 𝑓 +o 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) +o 𝑥)))), ∅) = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ ((𝑥 ∈ ((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) ↦ (dom 𝑓 +o 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) +o 𝑥)))), ∅)
6 eqid 2737 . 2 ((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) = ((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)))
7 eqid 2737 . 2 ((𝑥 ∈ ((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) ↦ (dom 𝑓 +o 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) +o 𝑥))) = ((𝑥 ∈ ((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) ↦ (dom 𝑓 +o 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) +o 𝑥)))
8 eqid 2737 . 2 (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))) = (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))
9 eqid 2737 . 2 (𝑢 ∈ (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))), 𝑣 ∈ (ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ↦ (((((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ·o 𝑣) +o 𝑢)) = (𝑢 ∈ (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))), 𝑣 ∈ (ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ↦ (((((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ·o 𝑣) +o 𝑢))
10 eqid 2737 . 2 (𝑢 ∈ (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))), 𝑣 ∈ (ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ·o 𝑢) +o 𝑣)) = (𝑢 ∈ (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))), 𝑣 ∈ (ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ·o 𝑢) +o 𝑣))
11 eqid 2737 . 2 (((𝑢 ∈ (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))), 𝑣 ∈ (ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ↦ (((((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ·o 𝑣) +o 𝑢)) ∘ (𝑢 ∈ (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))), 𝑣 ∈ (ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ·o 𝑢) +o 𝑣))) ∘ (seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ ((𝑥 ∈ ((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) ↦ (dom 𝑓 +o 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) +o 𝑥)))), ∅)‘dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) = (((𝑢 ∈ (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))), 𝑣 ∈ (ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ↦ (((((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ·o 𝑣) +o 𝑢)) ∘ (𝑢 ∈ (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))), 𝑣 ∈ (ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ·o 𝑢) +o 𝑣))) ∘ (seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ ((𝑥 ∈ ((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) ↦ (dom 𝑓 +o 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) +o 𝑥)))), ∅)‘dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))))
12 eqid 2737 . 2 (𝑏 ∈ (ω ↑o 𝐴) ↦ (((𝑢 ∈ (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))), 𝑣 ∈ (ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ↦ (((((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ·o 𝑣) +o 𝑢)) ∘ (𝑢 ∈ (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))), 𝑣 ∈ (ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ·o 𝑢) +o 𝑣))) ∘ (seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ ((𝑥 ∈ ((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) ↦ (dom 𝑓 +o 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) +o 𝑥)))), ∅)‘dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))))) = (𝑏 ∈ (ω ↑o 𝐴) ↦ (((𝑢 ∈ (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))), 𝑣 ∈ (ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ↦ (((((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ·o 𝑣) +o 𝑢)) ∘ (𝑢 ∈ (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))), 𝑣 ∈ (ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘ dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))) ·o 𝑢) +o 𝑣))) ∘ (seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ ((𝑥 ∈ ((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) ↦ (dom 𝑓 +o 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (((ω ↑o (OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘)) ·o (((ω CNF 𝐴)‘𝑏)‘(OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))‘𝑘))) +o 𝑥)))), ∅)‘dom OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)))))
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12cnfcom3clem 9745 1 (𝐴 ∈ On → ∃𝑔𝑏𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔𝑏):𝑏1-1-onto→(ω ↑o 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1779  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  cun 3949  wss 3951  c0 4333   cuni 4907  cmpt 5225   E cep 5583  ccnv 5684  dom cdm 5685  ccom 5689  Oncon0 6384  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433  ωcom 7887   supp csupp 8185  seqωcseqom 8487  1oc1o 8499   +o coa 8503   ·o comu 8504  o coe 8505  OrdIsocoi 9549   CNF ccnf 9701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-oexp 8512  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-cnf 9702
This theorem is referenced by:  infxpenc2  10062
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