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Theorem cnfcom3c 8887
Description: Wrap the construction of cnfcom3 8885 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 2825 . 2 dom (ω CNF 𝐴) = dom (ω CNF 𝐴)
2 eqid 2825 . 2 ((ω CNF 𝐴)‘𝑏) = ((ω CNF 𝐴)‘𝑏)
3 eqid 2825 . 2 OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)) = OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))
4 eqid 2825 . 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 2825 . 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 2825 . 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 2825 . 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 2825 . 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 2825 . 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 2825 . 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 2825 . 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 2825 . 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 8886 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 1878  wcel 2164  wral 3117  wrex 3118  Vcvv 3414  cdif 3795  cun 3796  wss 3798  c0 4146   cuni 4660  cmpt 4954   E cep 5256  ccnv 5345  dom cdm 5346  ccom 5350  Oncon0 5967  1-1-ontowf1o 6126  cfv 6127  (class class class)co 6910  cmpt2 6912  ωcom 7331   supp csupp 7564  seq𝜔cseqom 7813  1oc1o 7824   +o coa 7828   ·o comu 7829  o coe 7830  OrdIsocoi 8690   CNF ccnf 8842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-inf2 8822
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-se 5306  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-isom 6136  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-supp 7565  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-seqom 7814  df-1o 7831  df-2o 7832  df-oadd 7835  df-omul 7836  df-oexp 7837  df-er 8014  df-map 8129  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-fsupp 8551  df-oi 8691  df-cnf 8843
This theorem is referenced by:  infxpenc2  9165
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