Theorem cnfcom3c 9464

Description: Wrap the construction of cnfcom3 9462 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 → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔‘𝑏):𝑏–1-1-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.

Step	Hyp	Ref	Expression
1		eqid 2738	. 2 ⊢ dom (ω CNF 𝐴) = dom (ω CNF 𝐴)
2		eqid 2738	. 2 ⊢ (◡(ω CNF 𝐴)‘𝑏) = (◡(ω CNF 𝐴)‘𝑏)
3		eqid 2738	. 2 ⊢ OrdIso( E , ((◡(ω CNF 𝐴)‘𝑏) supp ∅)) = OrdIso( E , ((◡(ω CNF 𝐴)‘𝑏) supp ∅))
4		eqid 2738	. 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 2738	. 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 2738	. 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 2738	. 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 2738	. 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 2738	. 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 2738	. 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 2738	. 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 2738	. 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 ∅)))))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cnfcom3clem 9463	1 ⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))

Syntax hints:   → wi 4  ∃wex 1782   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  ∪ cuni 4839   ↦ cmpt 5157   E cep 5494  ^◡ccnv 5588  dom cdm 5589   ∘ ccom 5593  Oncon0 6266  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  ωcom 7712   supp csupp 7977  seq_ωcseqom 8278  1_oc1o 8290   +_o coa 8294   ·_o comu 8295   ↑_o coe 8296  OrdIsocoi 9268   CNF ccnf 9419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-seqom 8279  df-1o 8297  df-2o 8298  df-oadd 8301  df-omul 8302  df-oexp 8303  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-oi 9269  df-cnf 9420

This theorem is referenced by:  infxpenc2  9778