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Theorem satfv0fvfmla0 35418
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 35376 . . . 4 ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
2 satfv0fv.s . . . . . 6 𝑆 = (𝑀 Sat 𝐸)
32fveq1i 6907 . . . . 5 (𝑆‘∅) = ((𝑀 Sat 𝐸)‘∅)
43funeqi 6587 . . . 4 (Fun (𝑆‘∅) ↔ Fun ((𝑀 Sat 𝐸)‘∅))
51, 4sylibr 234 . . 3 ((𝑀𝑉𝐸𝑊) → Fun (𝑆‘∅))
653adant3 1133 . 2 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → Fun (𝑆‘∅))
7 fmla0 35387 . . . . . . . 8 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}
87eleq2i 2833 . . . . . . 7 (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)})
9 eqeq1 2741 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝑋 = (𝑖𝑔𝑗)))
1092rexbidv 3222 . . . . . . . 8 (𝑥 = 𝑋 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖𝑔𝑗)))
1110elrab 3692 . . . . . . 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 35352 . . . . . . . . . . . . . 14 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
1514eqeq2d 2748 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑋 = (𝑖𝑔𝑗) ↔ 𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
16 2fveq3 6911 . . . . . . . . . . . . . . . 16 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (1st ‘(2nd𝑋)) = (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)))
17 0ex 5307 . . . . . . . . . . . . . . . . . . 19 ∅ ∈ V
18 opex 5469 . . . . . . . . . . . . . . . . . . 19 𝑖, 𝑗⟩ ∈ V
1917, 18op2nd 8023 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩) = ⟨𝑖, 𝑗
2019fveq2i 6909 . . . . . . . . . . . . . . . . 17 (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = (1st ‘⟨𝑖, 𝑗⟩)
21 vex 3484 . . . . . . . . . . . . . . . . . 18 𝑖 ∈ V
22 vex 3484 . . . . . . . . . . . . . . . . . 18 𝑗 ∈ V
2321, 22op1st 8022 . . . . . . . . . . . . . . . . 17 (1st ‘⟨𝑖, 𝑗⟩) = 𝑖
2420, 23eqtri 2765 . . . . . . . . . . . . . . . 16 (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = 𝑖
2516, 24eqtrdi 2793 . . . . . . . . . . . . . . 15 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (1st ‘(2nd𝑋)) = 𝑖)
2625fveq2d 6910 . . . . . . . . . . . . . 14 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑎‘(1st ‘(2nd𝑋))) = (𝑎𝑖))
27 2fveq3 6911 . . . . . . . . . . . . . . . 16 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (2nd ‘(2nd𝑋)) = (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)))
2819fveq2i 6909 . . . . . . . . . . . . . . . . 17 (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = (2nd ‘⟨𝑖, 𝑗⟩)
2921, 22op2nd 8023 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨𝑖, 𝑗⟩) = 𝑗
3028, 29eqtri 2765 . . . . . . . . . . . . . . . 16 (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = 𝑗
3127, 30eqtrdi 2793 . . . . . . . . . . . . . . 15 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (2nd ‘(2nd𝑋)) = 𝑗)
3231fveq2d 6910 . . . . . . . . . . . . . 14 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑎‘(2nd ‘(2nd𝑋))) = (𝑎𝑗))
3326, 32breq12d 5156 . . . . . . . . . . . . 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 3444 . . . . . . . . . 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 3168 . . . . . . 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 3081 . . . . . 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 1136 . . . 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 1139 . . . . 5 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ (Fmla‘∅))
44 ovex 7464 . . . . . 6 (𝑀m ω) ∈ V
4544rabex 5339 . . . . 5 {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} ∈ V
46 eqeq1 2741 . . . . . . . 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 3222 . . . . . 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 5538 . . . . 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 35363 . . . . 5 ((𝑀𝑉𝐸𝑊) → (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})
5352eleq2d 2827 . . . 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 1133 . . 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 6958 . 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 1087   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  Vcvv 3480  c0 4333  cop 4632   class class class wbr 5143  {copab 5205  Fun wfun 6555  cfv 6561  (class class class)co 7431  ωcom 7887  1st c1st 8012  2nd c2nd 8013  m cmap 8866  𝑔cgoe 35338   Sat csat 35341  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-map 8868  df-goel 35345  df-sat 35348  df-fmla 35350
This theorem is referenced by:  satefvfmla0  35423
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