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Theorem cnfcom3c 9638
Description: Wrap the construction of cnfcom3 9636 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 2736 . 2 dom (ω CNF 𝐴) = dom (ω CNF 𝐴)
2 eqid 2736 . 2 ((ω CNF 𝐴)‘𝑏) = ((ω CNF 𝐴)‘𝑏)
3 eqid 2736 . 2 OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)) = OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))
4 eqid 2736 . 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 2736 . 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 2736 . 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 2736 . 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 2736 . 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 2736 . 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 2736 . 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 2736 . 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 2736 . 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 9637 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 2106  wral 3062  wrex 3071  Vcvv 3443  cdif 3905  cun 3906  wss 3908  c0 4280   cuni 4863  cmpt 5186   E cep 5534  ccnv 5630  dom cdm 5631  ccom 5635  Oncon0 6315  1-1-ontowf1o 6492  cfv 6493  (class class class)co 7353  cmpo 7355  ωcom 7798   supp csupp 8088  seqωcseqom 8389  1oc1o 8401   +o coa 8405   ·o comu 8406  o coe 8407  OrdIsocoi 9441   CNF ccnf 9593
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-inf2 9573
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-supp 8089  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-seqom 8390  df-1o 8408  df-2o 8409  df-oadd 8412  df-omul 8413  df-oexp 8414  df-er 8644  df-map 8763  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-fsupp 9302  df-oi 9442  df-cnf 9594
This theorem is referenced by:  infxpenc2  9954
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