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Theorem fmla0disjsuc 35383
Description: The set of valid Godel formulas of height 0 is disjoint with the formulas constructed from Godel-sets for the Sheffer stroke NAND and Godel-set of universal quantification. (Contributed by AV, 20-Oct-2023.)
Assertion
Ref Expression
fmla0disjsuc ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ∅
Distinct variable group:   𝑢,𝑖,𝑣,𝑥

Proof of Theorem fmla0disjsuc
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fmla0 35367 . . . 4 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘)}
2 rabab 3510 . . . 4 {𝑥 ∈ V ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘)} = {𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘)}
31, 2eqtri 2763 . . 3 (Fmla‘∅) = {𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘)}
43ineq1i 4224 . 2 ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ({𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘)} ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)})
5 inab 4315 . . 3 ({𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘)} ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = {𝑥 ∣ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘) ∧ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))}
6 goel 35332 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑗𝑔𝑘) = ⟨∅, ⟨𝑗, 𝑘⟩⟩)
76eqeq2d 2746 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗𝑔𝑘) ↔ 𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩))
8 1n0 8525 . . . . . . . . . . . . . . . . . . . 20 1o ≠ ∅
98nesymi 2996 . . . . . . . . . . . . . . . . . . 19 ¬ ∅ = 1o
109intnanr 487 . . . . . . . . . . . . . . . . . 18 ¬ (∅ = 1o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑢, 𝑣⟩)
11 gonafv 35335 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 ∈ V ∧ 𝑣 ∈ V) → (𝑢𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
1211el2v 3485 . . . . . . . . . . . . . . . . . . . 20 (𝑢𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩
1312eqeq2i 2748 . . . . . . . . . . . . . . . . . . 19 (⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢𝑔𝑣) ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
14 0ex 5313 . . . . . . . . . . . . . . . . . . . 20 ∅ ∈ V
15 opex 5475 . . . . . . . . . . . . . . . . . . . 20 𝑗, 𝑘⟩ ∈ V
1614, 15opth 5487 . . . . . . . . . . . . . . . . . . 19 (⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨1o, ⟨𝑢, 𝑣⟩⟩ ↔ (∅ = 1o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑢, 𝑣⟩))
1713, 16bitri 275 . . . . . . . . . . . . . . . . . 18 (⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢𝑔𝑣) ↔ (∅ = 1o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑢, 𝑣⟩))
1810, 17mtbir 323 . . . . . . . . . . . . . . . . 17 ¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢𝑔𝑣)
19 eqeq1 2739 . . . . . . . . . . . . . . . . 17 (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → (𝑥 = (𝑢𝑔𝑣) ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢𝑔𝑣)))
2018, 19mtbiri 327 . . . . . . . . . . . . . . . 16 (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → ¬ 𝑥 = (𝑢𝑔𝑣))
217, 20biimtrdi 253 . . . . . . . . . . . . . . 15 ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗𝑔𝑘) → ¬ 𝑥 = (𝑢𝑔𝑣)))
2221imp 406 . . . . . . . . . . . . . 14 (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗𝑔𝑘)) → ¬ 𝑥 = (𝑢𝑔𝑣))
2322adantr 480 . . . . . . . . . . . . 13 ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ¬ 𝑥 = (𝑢𝑔𝑣))
2423ralrimivw 3148 . . . . . . . . . . . 12 ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢𝑔𝑣))
25 2on0 8521 . . . . . . . . . . . . . . . . . . . . 21 2o ≠ ∅
2625nesymi 2996 . . . . . . . . . . . . . . . . . . . 20 ¬ ∅ = 2o
2726orci 865 . . . . . . . . . . . . . . . . . . 19 (¬ ∅ = 2o ∨ ¬ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩)
2814, 15opth 5487 . . . . . . . . . . . . . . . . . . . . 21 (⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ (∅ = 2o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
2928notbii 320 . . . . . . . . . . . . . . . . . . . 20 (¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ ¬ (∅ = 2o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
30 ianor 983 . . . . . . . . . . . . . . . . . . . 20 (¬ (∅ = 2o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩) ↔ (¬ ∅ = 2o ∨ ¬ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
3129, 30bitri 275 . . . . . . . . . . . . . . . . . . 19 (¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩ ↔ (¬ ∅ = 2o ∨ ¬ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩))
3227, 31mpbir 231 . . . . . . . . . . . . . . . . . 18 ¬ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩
33 eqeq1 2739 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → (𝑥 = ∀𝑔𝑖𝑢 ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ∀𝑔𝑖𝑢))
34 df-goal 35327 . . . . . . . . . . . . . . . . . . . 20 𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
3534eqeq2i 2748 . . . . . . . . . . . . . . . . . . 19 (⟨∅, ⟨𝑗, 𝑘⟩⟩ = ∀𝑔𝑖𝑢 ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩)
3633, 35bitrdi 287 . . . . . . . . . . . . . . . . . 18 (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → (𝑥 = ∀𝑔𝑖𝑢 ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨2o, ⟨𝑖, 𝑢⟩⟩))
3732, 36mtbiri 327 . . . . . . . . . . . . . . . . 17 (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → ¬ 𝑥 = ∀𝑔𝑖𝑢)
387, 37biimtrdi 253 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗𝑔𝑘) → ¬ 𝑥 = ∀𝑔𝑖𝑢))
3938imp 406 . . . . . . . . . . . . . . 15 (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗𝑔𝑘)) → ¬ 𝑥 = ∀𝑔𝑖𝑢)
4039adantr 480 . . . . . . . . . . . . . 14 ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ¬ 𝑥 = ∀𝑔𝑖𝑢)
4140adantr 480 . . . . . . . . . . . . 13 (((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) ∧ 𝑖 ∈ ω) → ¬ 𝑥 = ∀𝑔𝑖𝑢)
4241ralrimiva 3144 . . . . . . . . . . . 12 ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢)
4324, 42jca 511 . . . . . . . . . . 11 ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → (∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢))
4443ralrimiva 3144 . . . . . . . . . 10 (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗𝑔𝑘)) → ∀𝑢 ∈ (Fmla‘∅)(∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢))
45 ralnex 3070 . . . . . . . . . . . . . 14 (∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢𝑔𝑣) ↔ ¬ ∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣))
46 ralnex 3070 . . . . . . . . . . . . . 14 (∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢 ↔ ¬ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)
4745, 46anbi12i 628 . . . . . . . . . . . . 13 ((∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
48 ioran 985 . . . . . . . . . . . . 13 (¬ (∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ (¬ ∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∧ ¬ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
4947, 48bitr4i 278 . . . . . . . . . . . 12 ((∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢) ↔ ¬ (∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
5049ralbii 3091 . . . . . . . . . . 11 (∀𝑢 ∈ (Fmla‘∅)(∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢) ↔ ∀𝑢 ∈ (Fmla‘∅) ¬ (∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
51 ralnex 3070 . . . . . . . . . . 11 (∀𝑢 ∈ (Fmla‘∅) ¬ (∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢) ↔ ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
5250, 51bitri 275 . . . . . . . . . 10 (∀𝑢 ∈ (Fmla‘∅)(∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢) ↔ ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
5344, 52sylib 218 . . . . . . . . 9 (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗𝑔𝑘)) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
5453ex 412 . . . . . . . 8 ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
5554rexlimdva 3153 . . . . . . 7 (𝑗 ∈ ω → (∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
5655rexlimiv 3146 . . . . . 6 (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
5756imori 854 . . . . 5 (¬ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘) ∨ ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
58 ianor 983 . . . . 5 (¬ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘) ∧ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)) ↔ (¬ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘) ∨ ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
5957, 58mpbir 231 . . . 4 ¬ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘) ∧ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
6059abf 4412 . . 3 {𝑥 ∣ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘) ∧ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))} = ∅
615, 60eqtri 2763 . 2 ({𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘)} ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ∅
624, 61eqtri 2763 1 ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wo 847   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cin 3962  c0 4339  cop 4637  cfv 6563  (class class class)co 7431  ωcom 7887  1oc1o 8498  2oc2o 8499  𝑔cgoe 35318  𝑔cgna 35319  𝑔cgol 35320  Fmlacfmla 35322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-map 8867  df-goel 35325  df-gona 35326  df-goal 35327  df-sat 35328  df-fmla 35330
This theorem is referenced by:  satffunlem1lem2  35388
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