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Theorem satfv0fvfmla0 35398
Description: The value of the satisfaction predicate as function over a wff code at . (Contributed by AV, 2-Nov-2023.)
Hypothesis
Ref Expression
satfv0fv.s 𝑆 = (𝑀 Sat 𝐸)
Assertion
Ref Expression
satfv0fvfmla0 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → ((𝑆‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))})
Distinct variable groups:   𝐸,𝑎   𝑀,𝑎   𝑋,𝑎
Allowed substitution hints:   𝑆(𝑎)   𝑉(𝑎)   𝑊(𝑎)

Proof of Theorem satfv0fvfmla0
Dummy variables 𝑖 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 satfv0fun 35356 . . . 4 ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
2 satfv0fv.s . . . . . 6 𝑆 = (𝑀 Sat 𝐸)
32fveq1i 6908 . . . . 5 (𝑆‘∅) = ((𝑀 Sat 𝐸)‘∅)
43funeqi 6589 . . . 4 (Fun (𝑆‘∅) ↔ Fun ((𝑀 Sat 𝐸)‘∅))
51, 4sylibr 234 . . 3 ((𝑀𝑉𝐸𝑊) → Fun (𝑆‘∅))
653adant3 1131 . 2 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → Fun (𝑆‘∅))
7 fmla0 35367 . . . . . . . 8 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}
87eleq2i 2831 . . . . . . 7 (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)})
9 eqeq1 2739 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝑋 = (𝑖𝑔𝑗)))
1092rexbidv 3220 . . . . . . . 8 (𝑥 = 𝑋 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖𝑔𝑗)))
1110elrab 3695 . . . . . . 7 (𝑋 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)} ↔ (𝑋 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖𝑔𝑗)))
128, 11bitri 275 . . . . . 6 (𝑋 ∈ (Fmla‘∅) ↔ (𝑋 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖𝑔𝑗)))
13 simpr 484 . . . . . . . . . 10 (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖𝑔𝑗)) → 𝑋 = (𝑖𝑔𝑗))
14 goel 35332 . . . . . . . . . . . . . 14 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
1514eqeq2d 2746 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑋 = (𝑖𝑔𝑗) ↔ 𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
16 2fveq3 6912 . . . . . . . . . . . . . . . 16 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (1st ‘(2nd𝑋)) = (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)))
17 0ex 5313 . . . . . . . . . . . . . . . . . . 19 ∅ ∈ V
18 opex 5475 . . . . . . . . . . . . . . . . . . 19 𝑖, 𝑗⟩ ∈ V
1917, 18op2nd 8022 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩) = ⟨𝑖, 𝑗
2019fveq2i 6910 . . . . . . . . . . . . . . . . 17 (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = (1st ‘⟨𝑖, 𝑗⟩)
21 vex 3482 . . . . . . . . . . . . . . . . . 18 𝑖 ∈ V
22 vex 3482 . . . . . . . . . . . . . . . . . 18 𝑗 ∈ V
2321, 22op1st 8021 . . . . . . . . . . . . . . . . 17 (1st ‘⟨𝑖, 𝑗⟩) = 𝑖
2420, 23eqtri 2763 . . . . . . . . . . . . . . . 16 (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = 𝑖
2516, 24eqtrdi 2791 . . . . . . . . . . . . . . 15 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (1st ‘(2nd𝑋)) = 𝑖)
2625fveq2d 6911 . . . . . . . . . . . . . 14 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑎‘(1st ‘(2nd𝑋))) = (𝑎𝑖))
27 2fveq3 6912 . . . . . . . . . . . . . . . 16 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (2nd ‘(2nd𝑋)) = (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)))
2819fveq2i 6910 . . . . . . . . . . . . . . . . 17 (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = (2nd ‘⟨𝑖, 𝑗⟩)
2921, 22op2nd 8022 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨𝑖, 𝑗⟩) = 𝑗
3028, 29eqtri 2763 . . . . . . . . . . . . . . . 16 (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = 𝑗
3127, 30eqtrdi 2791 . . . . . . . . . . . . . . 15 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (2nd ‘(2nd𝑋)) = 𝑗)
3231fveq2d 6911 . . . . . . . . . . . . . 14 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑎‘(2nd ‘(2nd𝑋))) = (𝑎𝑗))
3326, 32breq12d 5161 . . . . . . . . . . . . 13 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → ((𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎𝑖)𝐸(𝑎𝑗)))
3415, 33biimtrdi 253 . . . . . . . . . . . 12 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑋 = (𝑖𝑔𝑗) → ((𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎𝑖)𝐸(𝑎𝑗))))
3534imp 406 . . . . . . . . . . 11 (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖𝑔𝑗)) → ((𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎𝑖)𝐸(𝑎𝑗)))
3635rabbidv 3441 . . . . . . . . . 10 (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖𝑔𝑗)) → {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})
3713, 36jca 511 . . . . . . . . 9 (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖𝑔𝑗)) → (𝑋 = (𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))
3837ex 412 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑋 = (𝑖𝑔𝑗) → (𝑋 = (𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})))
3938reximdva 3166 . . . . . . 7 (𝑖 ∈ ω → (∃𝑗 ∈ ω 𝑋 = (𝑖𝑔𝑗) → ∃𝑗 ∈ ω (𝑋 = (𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})))
4039reximia 3079 . . . . . 6 (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖𝑔𝑗) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))
4112, 40simplbiim 504 . . . . 5 (𝑋 ∈ (Fmla‘∅) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))
42413ad2ant3 1134 . . . 4 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))
43 simp3 1137 . . . . 5 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ (Fmla‘∅))
44 ovex 7464 . . . . . 6 (𝑀m ω) ∈ V
4544rabex 5345 . . . . 5 {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} ∈ V
46 eqeq1 2739 . . . . . . . 8 (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} ↔ {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))
479, 46bi2anan9 638 . . . . . . 7 ((𝑥 = 𝑋𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))}) → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ (𝑋 = (𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})))
48472rexbidv 3220 . . . . . 6 ((𝑥 = 𝑋𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})))
4948opelopabga 5543 . . . . 5 ((𝑋 ∈ (Fmla‘∅) ∧ {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} ∈ V) → (⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})} ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})))
5043, 45, 49sylancl 586 . . . 4 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → (⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})} ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖𝑔𝑗) ∧ {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})))
5142, 50mpbird 257 . . 3 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → ⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})
522satfv0 35343 . . . . 5 ((𝑀𝑉𝐸𝑊) → (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})
5352eleq2d 2825 . . . 4 ((𝑀𝑉𝐸𝑊) → (⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))}⟩ ∈ (𝑆‘∅) ↔ ⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})}))
54533adant3 1131 . . 3 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → (⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))}⟩ ∈ (𝑆‘∅) ↔ ⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})}))
5551, 54mpbird 257 . 2 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → ⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))}⟩ ∈ (𝑆‘∅))
56 funopfv 6959 . 2 (Fun (𝑆‘∅) → (⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))}⟩ ∈ (𝑆‘∅) → ((𝑆‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))}))
576, 55, 56sylc 65 1 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → ((𝑆‘∅)‘𝑋) = {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  Vcvv 3478  c0 4339  cop 4637   class class class wbr 5148  {copab 5210  Fun wfun 6557  cfv 6563  (class class class)co 7431  ωcom 7887  1st c1st 8011  2nd c2nd 8012  m cmap 8865  𝑔cgoe 35318   Sat csat 35321  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-map 8867  df-goel 35325  df-sat 35328  df-fmla 35330
This theorem is referenced by:  satefvfmla0  35403
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