Theorem cnfcom3c 9700

Description: Wrap the construction of cnfcom3 9698 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 2732	. 2 ⊢ dom (ω CNF 𝐴) = dom (ω CNF 𝐴)
2		eqid 2732	. 2 ⊢ (◡(ω CNF 𝐴)‘𝑏) = (◡(ω CNF 𝐴)‘𝑏)
3		eqid 2732	. 2 ⊢ OrdIso( E , ((◡(ω CNF 𝐴)‘𝑏) supp ∅)) = OrdIso( E , ((◡(ω CNF 𝐴)‘𝑏) supp ∅))
4		eqid 2732	. 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 2732	. 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 2732	. 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 2732	. 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 2732	. 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 2732	. 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 2732	. 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 2732	. 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 2732	. 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 9699	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 1781   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ∖ cdif 3945   ∪ cun 3946   ⊆ wss 3948  ∅c0 4322  ∪ cuni 4908   ↦ cmpt 5231   E cep 5579  ^◡ccnv 5675  dom cdm 5676   ∘ ccom 5680  Oncon0 6364  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7408   ∈ cmpo 7410  ωcom 7854   supp csupp 8145  seq_ωcseqom 8446  1_oc1o 8458   +_o coa 8462   ·_o comu 8463   ↑_o coe 8464  OrdIsocoi 9503   CNF ccnf 9655

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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-inf2 9635

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-supp 8146  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-seqom 8447  df-1o 8465  df-2o 8466  df-oadd 8469  df-omul 8470  df-oexp 8471  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-fsupp 9361  df-oi 9504  df-cnf 9656

This theorem is referenced by:  infxpenc2  10016