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Theorem satfv0 32840
 Description: The value of the satisfaction predicate as function over wff codes at ∅. (Contributed by AV, 8-Oct-2023.)
Hypothesis
Ref Expression
satfv0.s 𝑆 = (𝑀 Sat 𝐸)
Assertion
Ref Expression
satfv0 ((𝑀𝑉𝐸𝑊) → (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})
Distinct variable groups:   𝐸,𝑎,𝑖,𝑗,𝑥,𝑦   𝑀,𝑎,𝑖,𝑗,𝑥,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦,𝑖,𝑗,𝑎)   𝑉(𝑥,𝑦,𝑖,𝑗,𝑎)   𝑊(𝑥,𝑦,𝑖,𝑗,𝑎)

Proof of Theorem satfv0
Dummy variables 𝑓 𝑚 𝑛 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 peano1 7605 . . . 4 ∅ ∈ ω
2 elelsuc 6245 . . . 4 (∅ ∈ ω → ∅ ∈ suc ω)
31, 2mp1i 13 . . 3 ((𝑀𝑉𝐸𝑊) → ∅ ∈ suc ω)
4 satfv0.s . . . 4 𝑆 = (𝑀 Sat 𝐸)
54satfvsucom 32839 . . 3 ((𝑀𝑉𝐸𝑊 ∧ ∅ ∈ suc ω) → (𝑆‘∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝑓 (∃𝑣𝑓 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})‘∅))
63, 5mpd3an3 1459 . 2 ((𝑀𝑉𝐸𝑊) → (𝑆‘∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝑓 (∃𝑣𝑓 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})‘∅))
7 goelel3xp 32830 . . . . . . . . . 10 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) ∈ (ω × (ω × ω)))
8 eleq1 2839 . . . . . . . . . 10 (𝑥 = (𝑖𝑔𝑗) → (𝑥 ∈ (ω × (ω × ω)) ↔ (𝑖𝑔𝑗) ∈ (ω × (ω × ω))))
97, 8syl5ibrcom 250 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = (𝑖𝑔𝑗) → 𝑥 ∈ (ω × (ω × ω))))
109adantrd 495 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → 𝑥 ∈ (ω × (ω × ω))))
1110pm4.71d 565 . . . . . . 7 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ∧ 𝑥 ∈ (ω × (ω × ω)))))
12112rexbiia 3222 . . . . . 6 (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ∧ 𝑥 ∈ (ω × (ω × ω))))
13 r19.41vv 3267 . . . . . 6 (∃𝑖 ∈ ω ∃𝑗 ∈ ω ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ∧ 𝑥 ∈ (ω × (ω × ω))) ↔ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ∧ 𝑥 ∈ (ω × (ω × ω))))
14 ancom 464 . . . . . 6 ((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ∧ 𝑥 ∈ (ω × (ω × ω))) ↔ (𝑥 ∈ (ω × (ω × ω)) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})))
1512, 13, 143bitri 300 . . . . 5 (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ (𝑥 ∈ (ω × (ω × ω)) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})))
1615opabbii 5102 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (ω × (ω × ω)) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))}
17 omex 9144 . . . . 5 ω ∈ V
1817, 17xpex 7479 . . . . 5 (ω × ω) ∈ V
19 xpexg 7476 . . . . . 6 ((ω ∈ V ∧ (ω × ω) ∈ V) → (ω × (ω × ω)) ∈ V)
20 oveq1 7162 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → (𝑖𝑔𝑗) = (𝑚𝑔𝑗))
2120eqeq2d 2769 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑚𝑔𝑗)))
22 fveq2 6662 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑚 → (𝑎𝑖) = (𝑎𝑚))
2322breq1d 5045 . . . . . . . . . . . . . . 15 (𝑖 = 𝑚 → ((𝑎𝑖)𝐸(𝑎𝑗) ↔ (𝑎𝑚)𝐸(𝑎𝑗)))
2423rabbidv 3392 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)})
2524eqeq2d 2769 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} ↔ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)}))
2621, 25anbi12d 633 . . . . . . . . . . . 12 (𝑖 = 𝑚 → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ (𝑥 = (𝑚𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)})))
27 oveq2 7163 . . . . . . . . . . . . . 14 (𝑗 = 𝑛 → (𝑚𝑔𝑗) = (𝑚𝑔𝑛))
2827eqeq2d 2769 . . . . . . . . . . . . 13 (𝑗 = 𝑛 → (𝑥 = (𝑚𝑔𝑗) ↔ 𝑥 = (𝑚𝑔𝑛)))
29 fveq2 6662 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑛 → (𝑎𝑗) = (𝑎𝑛))
3029breq2d 5047 . . . . . . . . . . . . . . 15 (𝑗 = 𝑛 → ((𝑎𝑚)𝐸(𝑎𝑗) ↔ (𝑎𝑚)𝐸(𝑎𝑛)))
3130rabbidv 3392 . . . . . . . . . . . . . 14 (𝑗 = 𝑛 → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)})
3231eqeq2d 2769 . . . . . . . . . . . . 13 (𝑗 = 𝑛 → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)} ↔ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)}))
3328, 32anbi12d 633 . . . . . . . . . . . 12 (𝑗 = 𝑛 → ((𝑥 = (𝑚𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)}) ↔ (𝑥 = (𝑚𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)})))
3426, 33cbvrex2vw 3374 . . . . . . . . . . 11 (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ ∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑥 = (𝑚𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)}))
35 eqeq1 2762 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑖𝑔𝑗) → (𝑥 = (𝑚𝑔𝑛) ↔ (𝑖𝑔𝑗) = (𝑚𝑔𝑛)))
3635adantl 485 . . . . . . . . . . . . . . . . . 18 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖𝑔𝑗)) → (𝑥 = (𝑚𝑔𝑛) ↔ (𝑖𝑔𝑗) = (𝑚𝑔𝑛)))
37 goeleq12bg 32831 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑚𝑔𝑛) ↔ (𝑖 = 𝑚𝑗 = 𝑛)))
3822eqcomd 2764 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑚 → (𝑎𝑚) = (𝑎𝑖))
3929eqcomd 2764 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑛 → (𝑎𝑛) = (𝑎𝑗))
4038, 39breqan12d 5051 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 = 𝑚𝑗 = 𝑛) → ((𝑎𝑚)𝐸(𝑎𝑛) ↔ (𝑎𝑖)𝐸(𝑎𝑗)))
4140rabbidv 3392 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 = 𝑚𝑗 = 𝑛) → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})
4237, 41syl6bi 256 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑚𝑔𝑛) → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))
4342imp 410 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ (𝑖𝑔𝑗) = (𝑚𝑔𝑛)) → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})
44 eqeq12 2772 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → (𝑦 = 𝑧 ↔ {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))
4543, 44syl5ibrcom 250 . . . . . . . . . . . . . . . . . . . 20 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ (𝑖𝑔𝑗) = (𝑚𝑔𝑛)) → ((𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → 𝑦 = 𝑧))
4645exp4b 434 . . . . . . . . . . . . . . . . . . 19 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑚𝑔𝑛) → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} → (𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} → 𝑦 = 𝑧))))
4746adantr 484 . . . . . . . . . . . . . . . . . 18 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖𝑔𝑗)) → ((𝑖𝑔𝑗) = (𝑚𝑔𝑛) → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} → (𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} → 𝑦 = 𝑧))))
4836, 47sylbid 243 . . . . . . . . . . . . . . . . 17 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖𝑔𝑗)) → (𝑥 = (𝑚𝑔𝑛) → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} → (𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} → 𝑦 = 𝑧))))
4948impd 414 . . . . . . . . . . . . . . . 16 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖𝑔𝑗)) → ((𝑥 = (𝑚𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)}) → (𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} → 𝑦 = 𝑧)))
5049com23 86 . . . . . . . . . . . . . . 15 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖𝑔𝑗)) → (𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} → ((𝑥 = (𝑚𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)}) → 𝑦 = 𝑧)))
5150expimpd 457 . . . . . . . . . . . . . 14 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → ((𝑥 = (𝑚𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)}) → 𝑦 = 𝑧)))
5251rexlimdvva 3218 . . . . . . . . . . . . 13 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → ((𝑥 = (𝑚𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)}) → 𝑦 = 𝑧)))
5352com23 86 . . . . . . . . . . . 12 ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → ((𝑥 = (𝑚𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → 𝑦 = 𝑧)))
5453rexlimivv 3216 . . . . . . . . . . 11 (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑥 = (𝑚𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → 𝑦 = 𝑧))
5534, 54sylbi 220 . . . . . . . . . 10 (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → 𝑦 = 𝑧))
5655imp 410 . . . . . . . . 9 ((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})) → 𝑦 = 𝑧)
5756gen2 1798 . . . . . . . 8 𝑦𝑧((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})) → 𝑦 = 𝑧)
58 eqeq1 2762 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} ↔ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))
5958anbi2d 631 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})))
60592rexbidv 3224 . . . . . . . . 9 (𝑦 = 𝑧 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})))
6160mo4 2584 . . . . . . . 8 (∃*𝑦𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ ∀𝑦𝑧((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})) → 𝑦 = 𝑧))
6257, 61mpbir 234 . . . . . . 7 ∃*𝑦𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})
63 moabex 5322 . . . . . . 7 (∃*𝑦𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → {𝑦 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})} ∈ V)
6462, 63mp1i 13 . . . . . 6 (((ω ∈ V ∧ (ω × ω) ∈ V) ∧ 𝑥 ∈ (ω × (ω × ω))) → {𝑦 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})} ∈ V)
6519, 64opabex3d 7675 . . . . 5 ((ω ∈ V ∧ (ω × ω) ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (ω × (ω × ω)) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))} ∈ V)
6617, 18, 65mp2an 691 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (ω × (ω × ω)) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))} ∈ V
6716, 66eqeltri 2848 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})} ∈ V
6867rdg0 8072 . 2 (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝑓 (∃𝑣𝑓 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})}
696, 68eqtrdi 2809 1 ((𝑀𝑉𝐸𝑊) → (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∃*wmo 2555  {cab 2735  ∀wral 3070  ∃wrex 3071  {crab 3074  Vcvv 3409   ∖ cdif 3857   ∪ cun 3858   ∩ cin 3859  ∅c0 4227  {csn 4525  ⟨cop 4531   class class class wbr 5035  {copab 5097   ↦ cmpt 5115   × cxp 5525   ↾ cres 5529  suc csuc 6175  ‘cfv 6339  (class class class)co 7155  ωcom 7584  1st c1st 7696  2nd c2nd 7697  reccrdg 8060   ↑m cmap 8421  ∈𝑔cgoe 32815  ⊼𝑔cgna 32816  ∀𝑔cgol 32817   Sat csat 32818 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 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-inf2 9142 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-goel 32822  df-sat 32825 This theorem is referenced by:  satfv1  32845  satfrel  32849  satfdm  32851  satfrnmapom  32852  satfv0fun  32853  satfv0fvfmla0  32895
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