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 2761	. 2 ⊢ dom (ω CNF 𝐴) = dom (ω CNF 𝐴)
2		eqid 2761	. 2 ⊢ (◡(ω CNF 𝐴)‘𝑏) = (◡(ω CNF 𝐴)‘𝑏)
3		eqid 2761	. 2 ⊢ OrdIso( E , ((◡(ω CNF 𝐴)‘𝑏) supp ∅)) = OrdIso( E , ((◡(ω CNF 𝐴)‘𝑏) supp ∅))
4		eqid 2761	. 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 2761	. 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 2761	. 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 2761	. 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 2761	. 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 2761	. 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 2761	. 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 2761	. 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 2761	. 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)))

This proof depends on syntax axioms:   → wi 4  ∃wex 1812   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  ∪ cuni 4867   ↦ cmpt 5186   E cep 5550  ^◡ccnv 5650  dom cdm 5651   ∘ ccom 5655  Oncon0 6361  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  ωcom 7875   supp csupp 8170  seq_ωcseqom 8450  1_oc1o 8462   +_o coa 8466   ·_o comu 8467   ↑_o coe 8468  OrdIsocoi 9496   CNF ccnf 9655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-inf2 9635

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-supp 8171  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-seqom 8451  df-1o 8469  df-2o 8470  df-oadd 8473  df-omul 8474  df-oexp 8475  df-er 8710  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9347  df-oi 9497  df-cnf 9656

This theorem is used by:  infxpenc2  10094