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Theorem cnfcom3c 9157
 Description: Wrap the construction of cnfcom3 9155 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 2801 . 2 dom (ω CNF 𝐴) = dom (ω CNF 𝐴)
2 eqid 2801 . 2 ((ω CNF 𝐴)‘𝑏) = ((ω CNF 𝐴)‘𝑏)
3 eqid 2801 . 2 OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)) = OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))
4 eqid 2801 . 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 2801 . 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 2801 . 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 2801 . 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 2801 . 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 2801 . 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 2801 . 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 2801 . 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 2801 . 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 9156 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 1781   ∈ wcel 2112  ∀wral 3109  ∃wrex 3110  Vcvv 3444   ∖ cdif 3881   ∪ cun 3882   ⊆ wss 3884  ∅c0 4246  ∪ cuni 4803   ↦ cmpt 5113   E cep 5432  ◡ccnv 5522  dom cdm 5523   ∘ ccom 5527  Oncon0 6163  –1-1-onto→wf1o 6327  ‘cfv 6328  (class class class)co 7139   ∈ cmpo 7141  ωcom 7564   supp csupp 7817  seqωcseqom 8070  1oc1o 8082   +o coa 8086   ·o comu 8087   ↑o coe 8088  OrdIsocoi 8961   CNF ccnf 9112 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-seqom 8071  df-1o 8089  df-2o 8090  df-oadd 8093  df-omul 8094  df-oexp 8095  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-oi 8962  df-cnf 9113 This theorem is referenced by:  infxpenc2  9437
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