Theorem cnfcom3c 9394

Description: Wrap the construction of cnfcom3 9392 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 9393	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 1783   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836   ↦ cmpt 5153   E cep 5485  ^◡ccnv 5579  dom cdm 5580   ∘ ccom 5584  Oncon0 6251  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  ωcom 7687   supp csupp 7948  seq_ωcseqom 8248  1_oc1o 8260   +_o coa 8264   ·_o comu 8265   ↑_o coe 8266  OrdIsocoi 9198   CNF ccnf 9349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-seqom 8249  df-1o 8267  df-2o 8268  df-oadd 8271  df-omul 8272  df-oexp 8273  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-oi 9199  df-cnf 9350

This theorem is referenced by:  infxpenc2  9709