Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  satfv0fvfmla0 Structured version   Visualization version   GIF version

Theorem satfv0fvfmla0 32839
 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 32797 . . . 4 ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
2 satfv0fv.s . . . . . 6 𝑆 = (𝑀 Sat 𝐸)
32fveq1i 6656 . . . . 5 (𝑆‘∅) = ((𝑀 Sat 𝐸)‘∅)
43funeqi 6353 . . . 4 (Fun (𝑆‘∅) ↔ Fun ((𝑀 Sat 𝐸)‘∅))
51, 4sylibr 237 . . 3 ((𝑀𝑉𝐸𝑊) → Fun (𝑆‘∅))
653adant3 1129 . 2 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → Fun (𝑆‘∅))
7 fmla0 32808 . . . . . . . 8 (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)}
87eleq2i 2881 . . . . . . 7 (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)})
9 eqeq1 2802 . . . . . . . . 9 (𝑥 = 𝑋 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝑋 = (𝑖𝑔𝑗)))
1092rexbidv 3260 . . . . . . . 8 (𝑥 = 𝑋 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖𝑔𝑗)))
1110elrab 3630 . . . . . . 7 (𝑋 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖𝑔𝑗)} ↔ (𝑋 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖𝑔𝑗)))
128, 11bitri 278 . . . . . 6 (𝑋 ∈ (Fmla‘∅) ↔ (𝑋 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖𝑔𝑗)))
13 simpr 488 . . . . . . . . . 10 (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖𝑔𝑗)) → 𝑋 = (𝑖𝑔𝑗))
14 goel 32773 . . . . . . . . . . . . . 14 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
1514eqeq2d 2809 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑋 = (𝑖𝑔𝑗) ↔ 𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
16 2fveq3 6660 . . . . . . . . . . . . . . . 16 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (1st ‘(2nd𝑋)) = (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)))
17 0ex 5179 . . . . . . . . . . . . . . . . . . 19 ∅ ∈ V
18 opex 5325 . . . . . . . . . . . . . . . . . . 19 𝑖, 𝑗⟩ ∈ V
1917, 18op2nd 7693 . . . . . . . . . . . . . . . . . 18 (2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩) = ⟨𝑖, 𝑗
2019fveq2i 6658 . . . . . . . . . . . . . . . . 17 (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = (1st ‘⟨𝑖, 𝑗⟩)
21 vex 3445 . . . . . . . . . . . . . . . . . 18 𝑖 ∈ V
22 vex 3445 . . . . . . . . . . . . . . . . . 18 𝑗 ∈ V
2321, 22op1st 7692 . . . . . . . . . . . . . . . . 17 (1st ‘⟨𝑖, 𝑗⟩) = 𝑖
2420, 23eqtri 2821 . . . . . . . . . . . . . . . 16 (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = 𝑖
2516, 24eqtrdi 2849 . . . . . . . . . . . . . . 15 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (1st ‘(2nd𝑋)) = 𝑖)
2625fveq2d 6659 . . . . . . . . . . . . . 14 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑎‘(1st ‘(2nd𝑋))) = (𝑎𝑖))
27 2fveq3 6660 . . . . . . . . . . . . . . . 16 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (2nd ‘(2nd𝑋)) = (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)))
2819fveq2i 6658 . . . . . . . . . . . . . . . . 17 (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = (2nd ‘⟨𝑖, 𝑗⟩)
2921, 22op2nd 7693 . . . . . . . . . . . . . . . . 17 (2nd ‘⟨𝑖, 𝑗⟩) = 𝑗
3028, 29eqtri 2821 . . . . . . . . . . . . . . . 16 (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = 𝑗
3127, 30eqtrdi 2849 . . . . . . . . . . . . . . 15 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (2nd ‘(2nd𝑋)) = 𝑗)
3231fveq2d 6659 . . . . . . . . . . . . . 14 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑎‘(2nd ‘(2nd𝑋))) = (𝑎𝑗))
3326, 32breq12d 5047 . . . . . . . . . . . . 13 (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → ((𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎𝑖)𝐸(𝑎𝑗)))
3415, 33syl6bi 256 . . . . . . . . . . . 12 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑋 = (𝑖𝑔𝑗) → ((𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎𝑖)𝐸(𝑎𝑗))))
3534imp 410 . . . . . . . . . . 11 (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖𝑔𝑗)) → ((𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋))) ↔ (𝑎𝑖)𝐸(𝑎𝑗)))
3635rabbidv 3428 . . . . . . . . . 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 515 . . . . . . . . 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 416 . . . . . . . 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 3234 . . . . . . 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 3206 . . . . . 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 508 . . . . 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 1132 . . . 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 1135 . . . . 5 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ (Fmla‘∅))
44 ovex 7178 . . . . . 6 (𝑀m ω) ∈ V
4544rabex 5203 . . . . 5 {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))} ∈ V
46 eqeq1 2802 . . . . . . . 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 3260 . . . . . 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 5389 . . . . 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 589 . . . 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 260 . . 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 32784 . . . . 5 ((𝑀𝑉𝐸𝑊) → (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})
5352eleq2d 2875 . . . 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 1129 . . 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 260 . 2 ((𝑀𝑉𝐸𝑊𝑋 ∈ (Fmla‘∅)) → ⟨𝑋, {𝑎 ∈ (𝑀m ω) ∣ (𝑎‘(1st ‘(2nd𝑋)))𝐸(𝑎‘(2nd ‘(2nd𝑋)))}⟩ ∈ (𝑆‘∅))
56 funopfv 6702 . 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  {crab 3110  Vcvv 3442  ∅c0 4246  ⟨cop 4534   class class class wbr 5034  {copab 5096  Fun wfun 6326  ‘cfv 6332  (class class class)co 7145  ωcom 7573  1st c1st 7682  2nd c2nd 7683   ↑m cmap 8407  ∈𝑔cgoe 32759   Sat csat 32762  Fmlacfmla 32763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-inf2 9106 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-map 8409  df-goel 32766  df-sat 32769  df-fmla 32771 This theorem is referenced by:  satefvfmla0  32844
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