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Theorem fmla0disjsuc 35403
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 35387 . . . 4 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘)}
2 rabab 3512 . . . 4 {𝑥 ∈ V ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘)} = {𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘)}
31, 2eqtri 2765 . . 3 (Fmla‘∅) = {𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘)}
43ineq1i 4216 . 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 4309 . . 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 35352 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑗𝑔𝑘) = ⟨∅, ⟨𝑗, 𝑘⟩⟩)
76eqeq2d 2748 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ω ∧ 𝑘 ∈ ω) → (𝑥 = (𝑗𝑔𝑘) ↔ 𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩))
8 1n0 8526 . . . . . . . . . . . . . . . . . . . 20 1o ≠ ∅
98nesymi 2998 . . . . . . . . . . . . . . . . . . 19 ¬ ∅ = 1o
109intnanr 487 . . . . . . . . . . . . . . . . . 18 ¬ (∅ = 1o ∧ ⟨𝑗, 𝑘⟩ = ⟨𝑢, 𝑣⟩)
11 gonafv 35355 . . . . . . . . . . . . . . . . . . . . 21 ((𝑢 ∈ V ∧ 𝑣 ∈ V) → (𝑢𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
1211el2v 3487 . . . . . . . . . . . . . . . . . . . 20 (𝑢𝑔𝑣) = ⟨1o, ⟨𝑢, 𝑣⟩⟩
1312eqeq2i 2750 . . . . . . . . . . . . . . . . . . 19 (⟨∅, ⟨𝑗, 𝑘⟩⟩ = (𝑢𝑔𝑣) ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ⟨1o, ⟨𝑢, 𝑣⟩⟩)
14 0ex 5307 . . . . . . . . . . . . . . . . . . . 20 ∅ ∈ V
15 opex 5469 . . . . . . . . . . . . . . . . . . . 20 𝑗, 𝑘⟩ ∈ V
1614, 15opth 5481 . . . . . . . . . . . . . . . . . . 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 2741 . . . . . . . . . . . . . . . . 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 3150 . . . . . . . . . . . 12 ((((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗𝑔𝑘)) ∧ 𝑢 ∈ (Fmla‘∅)) → ∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢𝑔𝑣))
25 2on0 8522 . . . . . . . . . . . . . . . . . . . . 21 2o ≠ ∅
2625nesymi 2998 . . . . . . . . . . . . . . . . . . . 20 ¬ ∅ = 2o
2726orci 866 . . . . . . . . . . . . . . . . . . 19 (¬ ∅ = 2o ∨ ¬ ⟨𝑗, 𝑘⟩ = ⟨𝑖, 𝑢⟩)
2814, 15opth 5481 . . . . . . . . . . . . . . . . . . . . 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 984 . . . . . . . . . . . . . . . . . . . 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 2741 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ⟨∅, ⟨𝑗, 𝑘⟩⟩ → (𝑥 = ∀𝑔𝑖𝑢 ↔ ⟨∅, ⟨𝑗, 𝑘⟩⟩ = ∀𝑔𝑖𝑢))
34 df-goal 35347 . . . . . . . . . . . . . . . . . . . 20 𝑔𝑖𝑢 = ⟨2o, ⟨𝑖, 𝑢⟩⟩
3534eqeq2i 2750 . . . . . . . . . . . . . . . . . . 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 3146 . . . . . . . . . . . 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 3146 . . . . . . . . . 10 (((𝑗 ∈ ω ∧ 𝑘 ∈ ω) ∧ 𝑥 = (𝑗𝑔𝑘)) → ∀𝑢 ∈ (Fmla‘∅)(∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢))
45 ralnex 3072 . . . . . . . . . . . . . 14 (∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢𝑔𝑣) ↔ ¬ ∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣))
46 ralnex 3072 . . . . . . . . . . . . . 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 986 . . . . . . . . . . . . 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 3093 . . . . . . . . . . 11 (∀𝑢 ∈ (Fmla‘∅)(∀𝑣 ∈ (Fmla‘∅) ¬ 𝑥 = (𝑢𝑔𝑣) ∧ ∀𝑖 ∈ ω ¬ 𝑥 = ∀𝑔𝑖𝑢) ↔ ∀𝑢 ∈ (Fmla‘∅) ¬ (∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
51 ralnex 3072 . . . . . . . . . . 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 3155 . . . . . . 7 (𝑗 ∈ ω → (∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)))
5655rexlimiv 3148 . . . . . 6 (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘) → ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
5756imori 855 . . . . 5 (¬ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘) ∨ ¬ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))
58 ianor 984 . . . . 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 4406 . . 3 {𝑥 ∣ (∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘) ∧ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢))} = ∅
615, 60eqtri 2765 . 2 ({𝑥 ∣ ∃𝑗 ∈ ω ∃𝑘 ∈ ω 𝑥 = (𝑗𝑔𝑘)} ∩ {𝑥 ∣ ∃𝑢 ∈ (Fmla‘∅)(∃𝑣 ∈ (Fmla‘∅)𝑥 = (𝑢𝑔𝑣) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑢)}) = ∅
624, 61eqtri 2765 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 848   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cin 3950  c0 4333  cop 4632  cfv 6561  (class class class)co 7431  ωcom 7887  1oc1o 8499  2oc2o 8500  𝑔cgoe 35338  𝑔cgna 35339  𝑔cgol 35340  Fmlacfmla 35342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-map 8868  df-goel 35345  df-gona 35346  df-goal 35347  df-sat 35348  df-fmla 35350
This theorem is referenced by:  satffunlem1lem2  35408
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