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Theorem cnfcom3c 9744
Description: Wrap the construction of cnfcom3 9742 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 2735 . 2 dom (ω CNF 𝐴) = dom (ω CNF 𝐴)
2 eqid 2735 . 2 ((ω CNF 𝐴)‘𝑏) = ((ω CNF 𝐴)‘𝑏)
3 eqid 2735 . 2 OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅)) = OrdIso( E , (((ω CNF 𝐴)‘𝑏) supp ∅))
4 eqid 2735 . 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 2735 . 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 2735 . 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 2735 . 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 2735 . 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 2735 . 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 2735 . 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 2735 . 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 2735 . 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 9743 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 1776  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  cun 3961  wss 3963  c0 4339   cuni 4912  cmpt 5231   E cep 5588  ccnv 5688  dom cdm 5689  ccom 5693  Oncon0 6386  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  cmpo 7433  ωcom 7887   supp csupp 8184  seqωcseqom 8486  1oc1o 8498   +o coa 8502   ·o comu 8503  o coe 8504  OrdIsocoi 9547   CNF ccnf 9699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-oi 9548  df-cnf 9700
This theorem is referenced by:  infxpenc2  10060
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