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Theorem cantnfres 8571
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 7316 . . . . . . . . . . . 12 (𝜑 → ((𝑛𝐵𝑋) supp ∅) = ((𝑛𝐷𝑋) supp ∅))
5 oieq2 8415 . . . . . . . . . . . 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 6191 . . . . . . . . . 10 (𝜑 → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))
873ad2ant1 1081 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) = (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘))
98oveq2d 6663 . . . . . . . 8 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (𝐴𝑜 (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = (𝐴𝑜 (OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
10 suppssdm 7305 . . . . . . . . . . . . 13 ((𝑛𝐵𝑋) supp ∅) ⊆ dom (𝑛𝐵𝑋)
11 eqid 2621 . . . . . . . . . . . . . . 15 (𝑛𝐵𝑋) = (𝑛𝐵𝑋)
1211dmmptss 5629 . . . . . . . . . . . . . 14 dom (𝑛𝐵𝑋) ⊆ 𝐵
1312a1i 11 . . . . . . . . . . . . 13 (𝜑 → dom (𝑛𝐵𝑋) ⊆ 𝐵)
1410, 13syl5ss 3612 . . . . . . . . . . . 12 (𝜑 → ((𝑛𝐵𝑋) supp ∅) ⊆ 𝐵)
15143ad2ant1 1081 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐵𝑋) supp ∅) ⊆ 𝐵)
16 eqid 2621 . . . . . . . . . . . . . 14 OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐵𝑋) supp ∅))
1716oif 8432 . . . . . . . . . . . . 13 OrdIso( E , ((𝑛𝐵𝑋) supp ∅)):dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅))⟶((𝑛𝐵𝑋) supp ∅)
1817ffvelrni 6356 . . . . . . . . . . . 12 (𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ ((𝑛𝐵𝑋) supp ∅))
19183ad2ant2 1082 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ ((𝑛𝐵𝑋) supp ∅))
2015, 19sseldd 3602 . . . . . . . . . 10 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘) ∈ 𝐵)
21 fvres 6205 . . . . . . . . . 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 1081 . . . . . . . . . . 11 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → 𝐵𝐷)
2423resmptd 5450 . . . . . . . . . 10 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐷𝑋) ↾ 𝐵) = (𝑛𝐵𝑋))
2524fveq1d 6191 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → (((𝑛𝐷𝑋) ↾ 𝐵)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)))
268fveq2d 6193 . . . . . . . . 9 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
2722, 25, 263eqtr3d 2663 . . . . . . . 8 ((𝜑𝑘 ∈ dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) ∧ 𝑧 ∈ On) → ((𝑛𝐵𝑋)‘(OrdIso( E , ((𝑛𝐵𝑋) supp ∅))‘𝑘)) = ((𝑛𝐷𝑋)‘(OrdIso( E , ((𝑛𝐷𝑋) supp ∅))‘𝑘)))
289, 27oveq12d 6665 . . . . . . 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 6662 . . . . . 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 6716 . . . . 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 5324 . . . . . 6 (𝜑 → dom OrdIso( E , ((𝑛𝐵𝑋) supp ∅)) = dom OrdIso( E , ((𝑛𝐷𝑋) supp ∅)))
32 eqid 2621 . . . . . 6 On = On
33 mpt2eq12 6712 . . . . . 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 694 . . . . 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 2655 . . . 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 2621 . . . 4 ∅ = ∅
37 seqomeq12 7546 . . . 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 694 . . 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 6195 . 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 2621 . . 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 8563 . 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 2621 . . 3 OrdIso( E , ((𝑛𝐷𝑋) supp ∅)) = OrdIso( E , ((𝑛𝐷𝑋) supp ∅))
48 cantnfrescl.a . . . . 5 (𝜑 → ∅ ∈ 𝐴)
4940, 41, 42, 1, 2, 3, 48, 46cantnfrescl 8570 . . . 4 (𝜑 → ((𝑛𝐵𝑋) ∈ 𝑆 ↔ (𝑛𝐷𝑋) ∈ 𝑇))
5043, 49mpbid 222 . . 3 (𝜑 → (𝑛𝐷𝑋) ∈ 𝑇)
51 eqid 2621 . . 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 8563 . 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 2665 1 (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛𝐵𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛𝐷𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989  Vcvv 3198  cdif 3569  wss 3572  c0 3913  cmpt 4727   E cep 5026  dom cdm 5112  cres 5114  Oncon0 5721  cfv 5886  (class class class)co 6647  cmpt2 6649   supp csupp 7292  seq𝜔cseqom 7539   +𝑜 coa 7554   ·𝑜 comu 7555  𝑜 coe 7556  OrdIsocoi 8411   CNF ccnf 8555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-supp 7293  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-seqom 7540  df-oadd 7561  df-er 7739  df-map 7856  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-fsupp 8273  df-oi 8412  df-cnf 8556
This theorem is referenced by: (None)
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