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Theorem cantnfres 8457
 Description: The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnfs.s 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
cantnfrescl.d (𝜑𝐷 ∈ On)
cantnfrescl.b (𝜑𝐵𝐷)
cantnfrescl.x ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)
cantnfrescl.a (𝜑 → ∅ ∈ 𝐴)
cantnfrescl.t 𝑇 = dom (𝐴 CNF 𝐷)
cantnfres.m (𝜑 → (𝑛𝐵𝑋) ∈ 𝑆)
Assertion
Ref Expression
cantnfres (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)))
Distinct variable groups:   𝐵,𝑛   𝐷,𝑛   𝐴,𝑛   𝜑,𝑛
Allowed substitution hints:   𝑆(𝑛)   𝑇(𝑛)   𝑋(𝑛)

Proof of Theorem cantnfres
Dummy variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfrescl.d . . . . . . . . . . . . 13 (𝜑𝐷 ∈ On)
2 cantnfrescl.b . . . . . . . . . . . . 13 (𝜑𝐵𝐷)
3 cantnfrescl.x . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (𝐷𝐵)) → 𝑋 = ∅)
41, 2, 3extmptsuppeq 7206 . . . . . . . . . . . 12 (𝜑 → ((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅))
5 oieq2 8301 . . . . . . . . . . . 12 (((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅) → OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐷𝑋) supp ∅)))
64, 5syl 17 . . . . . . . . . . 11 (𝜑 → OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐷𝑋) supp ∅)))
76fveq1d 6105 . . . . . . . . . 10 (𝜑 → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))
873ad2ant1 1075 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))
98oveq2d 6565 . . . . . . . 8 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (𝐴𝑜 (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = (𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
10 suppssdm 7195 . . . . . . . . . . . . 13 ((𝑛𝐵𝑋) supp ∅) ⊆ dom (𝑛𝐵𝑋)
11 eqid 2610 . . . . . . . . . . . . . . 15 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
1211dmmptss 5548 . . . . . . . . . . . . . 14 dom (𝑛𝐵𝑋) ⊆ 𝐵
1312a1i 11 . . . . . . . . . . . . 13 (𝜑 → dom (𝑛𝐵𝑋) ⊆ 𝐵)
1410, 13syl5ss 3579 . . . . . . . . . . . 12 (𝜑 → ((𝑛𝐵𝑋) supp ∅) ⊆ 𝐵)
15143ad2ant1 1075 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐵𝑋) supp ∅) ⊆ 𝐵)
16 eqid 2610 . . . . . . . . . . . . . 14 OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐵𝑋) supp ∅))
1716oif 8318 . . . . . . . . . . . . 13 OrdIso( E , ((𝑛𝐵𝑋) supp ∅)):dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅))⟶((𝑛𝐵𝑋) supp ∅)
1817ffvelrni 6266 . . . . . . . . . . . 12 (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ ((𝑛𝐵𝑋) supp ∅))
19183ad2ant2 1076 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ ((𝑛𝐵𝑋) supp ∅))
2015, 19sseldd 3569 . . . . . . . . . 10 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ 𝐵)
21 fvres 6117 . . . . . . . . . 10 ((OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ 𝐵 → (((𝑛𝐷𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)))
2220, 21syl 17 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛𝐷𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)))
2323ad2ant1 1075 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → 𝐵𝐷)
2423resmptd 5371 . . . . . . . . . 10 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐷𝑋) ↾ 𝐵) = (𝑛𝐵𝑋))
2524fveq1d 6105 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛𝐷𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)))
268fveq2d 6107 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
2722, 25, 263eqtr3d 2652 . . . . . . . 8 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
289, 27oveq12d 6567 . . . . . . 7 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝐴𝑜 (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) = ((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))))
2928oveq1d 6564 . . . . . 6 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝐴𝑜 (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +𝑜 𝑧) = (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧))
3029mpt2eq3dva 6617 . . . . 5 (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)))
316dmeqd 5248 . . . . . 6 (𝜑 → dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)))
32 eqid 2610 . . . . . 6 On = On
33 mpt2eq12 6613 . . . . . 6 ((dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)) ∧ On = On) → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)))
3431, 32, 33sylancl 693 . . . . 5 (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)))
3530, 34eqtrd 2644 . . . 4 (𝜑 → (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)))
36 eqid 2610 . . . 4 ∅ = ∅
37 seqomeq12 7436 . . . 4 (((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)) = (𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)) ∧ ∅ = ∅) → seq𝜔((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)), ∅))
3835, 36, 37sylancl 693 . . 3 (𝜑 → seq𝜔((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)), ∅))
3938, 31fveq12d 6109 . 2 (𝜑 → (seq𝜔((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅))) = (seq𝜔((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅))))
40 cantnfs.s . . 3 𝑆 = dom (𝐴 CNF 𝐵)
41 cantnfs.a . . 3 (𝜑𝐴 ∈ On)
42 cantnfs.b . . 3 (𝜑𝐵 ∈ On)
43 cantnfres.m . . 3 (𝜑 → (𝑛𝐵𝑋) ∈ 𝑆)
44 eqid 2610 . . 3 seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)), ∅)
4540, 41, 42, 16, 43, 44cantnfval2 8449 . 2 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = (seq𝜔((𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅))))
46 cantnfrescl.t . . 3 𝑇 = dom (𝐴 CNF 𝐷)
47 eqid 2610 . . 3 OrdIso( E , ((𝑛𝐷𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐷𝑋) supp ∅))
48 cantnfrescl.a . . . . 5 (𝜑 → ∅ ∈ 𝐴)
4940, 41, 42, 1, 2, 3, 48, 46cantnfrescl 8456 . . . 4 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ (𝑛𝐷𝑋) ∈ 𝑇))
5043, 49mpbid 221 . . 3 (𝜑 → (𝑛𝐷𝑋) ∈ 𝑇)
51 eqid 2610 . . 3 seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)), ∅)
5246, 41, 1, 47, 50, 51cantnfval2 8449 . 2 (𝜑 → ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)) = (seq𝜔((𝑘 ∈ dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)), 𝑧 ∈ On ↦ (((𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)) ·𝑜 ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))) +𝑜 𝑧)), ∅)‘dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅))))
5339, 45, 523eqtr4d 2654 1 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874   ↦ cmpt 4643   E cep 4947  dom cdm 5038   ↾ cres 5040  Oncon0 5640  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   supp csupp 7182  seq𝜔cseqom 7429   +𝑜 coa 7444   ·𝑜 comu 7445   ↑𝑜 coe 7446  OrdIsocoi 8297   CNF ccnf 8441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-cnf 8442 This theorem is referenced by: (None)
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