Theorem cnfcom3c 9720

Description: Wrap the construction of cnfcom3 9718 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 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 ∅)))))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cnfcom3clem 9719	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 1779   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  Vcvv 3459   ∖ cdif 3923   ∪ cun 3924   ⊆ wss 3926  ∅c0 4308  ∪ cuni 4883   ↦ cmpt 5201   E cep 5552  ^◡ccnv 5653  dom cdm 5654   ∘ ccom 5658  Oncon0 6352  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  ωcom 7861   supp csupp 8159  seq_ωcseqom 8461  1_oc1o 8473   +_o coa 8477   ·_o comu 8478   ↑_o coe 8479  OrdIsocoi 9523   CNF ccnf 9675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-seqom 8462  df-1o 8480  df-2o 8481  df-oadd 8484  df-omul 8485  df-oexp 8486  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9374  df-oi 9524  df-cnf 9676

This theorem is referenced by:  infxpenc2  10036