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Theorem satfv0 35364
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 7911 . . . 4 ∅ ∈ ω
2 elelsuc 6456 . . . 4 (∅ ∈ ω → ∅ ∈ suc ω)
31, 2mp1i 13 . . 3 ((𝑀𝑉𝐸𝑊) → ∅ ∈ suc ω)
4 satfv0.s . . . 4 𝑆 = (𝑀 Sat 𝐸)
54satfvsucom 35363 . . 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 1463 . 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 35354 . . . . . . . . . 10 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) ∈ (ω × (ω × ω)))
8 eleq1 2828 . . . . . . . . . 10 (𝑥 = (𝑖𝑔𝑗) → (𝑥 ∈ (ω × (ω × ω)) ↔ (𝑖𝑔𝑗) ∈ (ω × (ω × ω))))
97, 8syl5ibrcom 247 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = (𝑖𝑔𝑗) → 𝑥 ∈ (ω × (ω × ω))))
109adantrd 491 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → 𝑥 ∈ (ω × (ω × ω))))
1110pm4.71d 561 . . . . . . 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 3217 . . . . . 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 3226 . . . . . 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 460 . . . . . 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 297 . . . . 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 5209 . . . 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 9684 . . . . 5 ω ∈ V
1817, 17xpex 7774 . . . . 5 (ω × ω) ∈ V
19 xpexg 7771 . . . . . 6 ((ω ∈ V ∧ (ω × ω) ∈ V) → (ω × (ω × ω)) ∈ V)
20 oveq1 7439 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → (𝑖𝑔𝑗) = (𝑚𝑔𝑗))
2120eqeq2d 2747 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑚𝑔𝑗)))
22 fveq2 6905 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑚 → (𝑎𝑖) = (𝑎𝑚))
2322breq1d 5152 . . . . . . . . . . . . . . 15 (𝑖 = 𝑚 → ((𝑎𝑖)𝐸(𝑎𝑗) ↔ (𝑎𝑚)𝐸(𝑎𝑗)))
2423rabbidv 3443 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)})
2524eqeq2d 2747 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} ↔ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)}))
2621, 25anbi12d 632 . . . . . . . . . . . 12 (𝑖 = 𝑚 → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ (𝑥 = (𝑚𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)})))
27 oveq2 7440 . . . . . . . . . . . . . 14 (𝑗 = 𝑛 → (𝑚𝑔𝑗) = (𝑚𝑔𝑛))
2827eqeq2d 2747 . . . . . . . . . . . . 13 (𝑗 = 𝑛 → (𝑥 = (𝑚𝑔𝑗) ↔ 𝑥 = (𝑚𝑔𝑛)))
29 fveq2 6905 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑛 → (𝑎𝑗) = (𝑎𝑛))
3029breq2d 5154 . . . . . . . . . . . . . . 15 (𝑗 = 𝑛 → ((𝑎𝑚)𝐸(𝑎𝑗) ↔ (𝑎𝑚)𝐸(𝑎𝑛)))
3130rabbidv 3443 . . . . . . . . . . . . . 14 (𝑗 = 𝑛 → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)})
3231eqeq2d 2747 . . . . . . . . . . . . 13 (𝑗 = 𝑛 → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)} ↔ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)}))
3328, 32anbi12d 632 . . . . . . . . . . . 12 (𝑗 = 𝑛 → ((𝑥 = (𝑚𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)}) ↔ (𝑥 = (𝑚𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)})))
3426, 33cbvrex2vw 3241 . . . . . . . . . . 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 2740 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑖𝑔𝑗) → (𝑥 = (𝑚𝑔𝑛) ↔ (𝑖𝑔𝑗) = (𝑚𝑔𝑛)))
3635adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖𝑔𝑗)) → (𝑥 = (𝑚𝑔𝑛) ↔ (𝑖𝑔𝑗) = (𝑚𝑔𝑛)))
37 goeleq12bg 35355 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑚𝑔𝑛) ↔ (𝑖 = 𝑚𝑗 = 𝑛)))
3822eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑚 → (𝑎𝑚) = (𝑎𝑖))
3929eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑛 → (𝑎𝑛) = (𝑎𝑗))
4038, 39breqan12d 5158 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 = 𝑚𝑗 = 𝑛) → ((𝑎𝑚)𝐸(𝑎𝑛) ↔ (𝑎𝑖)𝐸(𝑎𝑗)))
4140rabbidv 3443 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 = 𝑚𝑗 = 𝑛) → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})
4237, 41biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑚𝑔𝑛) → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))
4342imp 406 . . . . . . . . . . . . . . . . . . . . 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 2753 . . . . . . . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . . . . . . . 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 430 . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 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 240 . . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . 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 3212 . . . . . . . . . . . . 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 3200 . . . . . . . . . . 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 217 . . . . . . . . . 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 406 . . . . . . . . 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 1795 . . . . . . . 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 2740 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} ↔ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))
5958anbi2d 630 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})))
60592rexbidv 3221 . . . . . . . . 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 2565 . . . . . . . 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 231 . . . . . . 7 ∃*𝑦𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})
63 moabex 5463 . . . . . . 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 7991 . . . . 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 692 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (ω × (ω × ω)) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))} ∈ V
6716, 66eqeltri 2836 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})} ∈ V
6867rdg0 8462 . 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 2792 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 206  wa 395  wo 847  wal 1537   = wceq 1539  wcel 2107  ∃*wmo 2537  {cab 2713  wral 3060  wrex 3069  {crab 3435  Vcvv 3479  cdif 3947  cun 3948  cin 3949  c0 4332  {csn 4625  cop 4631   class class class wbr 5142  {copab 5204  cmpt 5224   × cxp 5682  cres 5686  suc csuc 6385  cfv 6560  (class class class)co 7432  ωcom 7888  1st c1st 8013  2nd c2nd 8014  reccrdg 8450  m cmap 8867  𝑔cgoe 35339  𝑔cgna 35340  𝑔cgol 35341   Sat csat 35342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-goel 35346  df-sat 35349
This theorem is referenced by:  satfv1  35369  satfrel  35373  satfdm  35375  satfrnmapom  35376  satfv0fun  35377  satfv0fvfmla0  35419
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