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Theorem satfv0 32491
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 7592 . . . 4 ∅ ∈ ω
2 elelsuc 6260 . . . 4 (∅ ∈ ω → ∅ ∈ suc ω)
31, 2mp1i 13 . . 3 ((𝑀𝑉𝐸𝑊) → ∅ ∈ suc ω)
4 satfv0.s . . . 4 𝑆 = (𝑀 Sat 𝐸)
54satfvsucom 32490 . . 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 1455 . 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 32481 . . . . . . . . . 10 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) ∈ (ω × (ω × ω)))
8 eleq1 2904 . . . . . . . . . 10 (𝑥 = (𝑖𝑔𝑗) → (𝑥 ∈ (ω × (ω × ω)) ↔ (𝑖𝑔𝑗) ∈ (ω × (ω × ω))))
97, 8syl5ibrcom 248 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = (𝑖𝑔𝑗) → 𝑥 ∈ (ω × (ω × ω))))
109adantrd 492 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → 𝑥 ∈ (ω × (ω × ω))))
1110pm4.71d 562 . . . . . . 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 3302 . . . . . 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 3353 . . . . . 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 461 . . . . . 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 298 . . . . 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 5129 . . . 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 9098 . . . . 5 ω ∈ V
1817, 17xpex 7468 . . . . 5 (ω × ω) ∈ V
19 xpexg 7465 . . . . . 6 ((ω ∈ V ∧ (ω × ω) ∈ V) → (ω × (ω × ω)) ∈ V)
20 oveq1 7158 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → (𝑖𝑔𝑗) = (𝑚𝑔𝑗))
2120eqeq2d 2836 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑚𝑔𝑗)))
22 fveq2 6666 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑚 → (𝑎𝑖) = (𝑎𝑚))
2322breq1d 5072 . . . . . . . . . . . . . . 15 (𝑖 = 𝑚 → ((𝑎𝑖)𝐸(𝑎𝑗) ↔ (𝑎𝑚)𝐸(𝑎𝑗)))
2423rabbidv 3485 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)})
2524eqeq2d 2836 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} ↔ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)}))
2621, 25anbi12d 630 . . . . . . . . . . . 12 (𝑖 = 𝑚 → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ (𝑥 = (𝑚𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)})))
27 oveq2 7159 . . . . . . . . . . . . . 14 (𝑗 = 𝑛 → (𝑚𝑔𝑗) = (𝑚𝑔𝑛))
2827eqeq2d 2836 . . . . . . . . . . . . 13 (𝑗 = 𝑛 → (𝑥 = (𝑚𝑔𝑗) ↔ 𝑥 = (𝑚𝑔𝑛)))
29 fveq2 6666 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑛 → (𝑎𝑗) = (𝑎𝑛))
3029breq2d 5074 . . . . . . . . . . . . . . 15 (𝑗 = 𝑛 → ((𝑎𝑚)𝐸(𝑎𝑗) ↔ (𝑎𝑚)𝐸(𝑎𝑛)))
3130rabbidv 3485 . . . . . . . . . . . . . 14 (𝑗 = 𝑛 → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)})
3231eqeq2d 2836 . . . . . . . . . . . . 13 (𝑗 = 𝑛 → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)} ↔ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)}))
3328, 32anbi12d 630 . . . . . . . . . . . 12 (𝑗 = 𝑛 → ((𝑥 = (𝑚𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)}) ↔ (𝑥 = (𝑚𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)})))
3426, 33cbvrex2v 3470 . . . . . . . . . . 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 2829 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑖𝑔𝑗) → (𝑥 = (𝑚𝑔𝑛) ↔ (𝑖𝑔𝑗) = (𝑚𝑔𝑛)))
3635adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖𝑔𝑗)) → (𝑥 = (𝑚𝑔𝑛) ↔ (𝑖𝑔𝑗) = (𝑚𝑔𝑛)))
37 goeleq12bg 32482 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑚𝑔𝑛) ↔ (𝑖 = 𝑚𝑗 = 𝑛)))
3822eqcomd 2831 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑚 → (𝑎𝑚) = (𝑎𝑖))
3929eqcomd 2831 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑛 → (𝑎𝑛) = (𝑎𝑗))
4038, 39breqan12d 5078 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 = 𝑚𝑗 = 𝑛) → ((𝑎𝑚)𝐸(𝑎𝑛) ↔ (𝑎𝑖)𝐸(𝑎𝑗)))
4140rabbidv 3485 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 = 𝑚𝑗 = 𝑛) → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})
4237, 41syl6bi 254 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑚𝑔𝑛) → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))
4342imp 407 . . . . . . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . . . . . . . . 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 248 . . . . . . . . . . . . . . . . . . . 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 431 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 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 241 . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . 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 3298 . . . . . . . . . . . . 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 3296 . . . . . . . . . . 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 218 . . . . . . . . . 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 407 . . . . . . . . 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 1790 . . . . . . . 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 2829 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} ↔ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))
5958anbi2d 628 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) ↔ (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})))
60592rexbidv 3304 . . . . . . . . 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 2647 . . . . . . . 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 232 . . . . . . 7 ∃*𝑦𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})
63 moabex 5347 . . . . . . 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 7660 . . . . 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 688 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (ω × (ω × ω)) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))} ∈ V
6716, 66eqeltri 2913 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})} ∈ V
6867rdg0 8051 . 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, 68syl6eq 2876 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 207  wa 396  wo 843  wal 1528   = wceq 1530  wcel 2107  ∃*wmo 2617  {cab 2803  wral 3142  wrex 3143  {crab 3146  Vcvv 3499  cdif 3936  cun 3937  cin 3938  c0 4294  {csn 4563  cop 4569   class class class wbr 5062  {copab 5124  cmpt 5142   × cxp 5551  cres 5555  suc csuc 6190  cfv 6351  (class class class)co 7151  ωcom 7571  1st c1st 7681  2nd c2nd 7682  reccrdg 8039  m cmap 8399  𝑔cgoe 32466  𝑔cgna 32467  𝑔cgol 32468   Sat csat 32469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-goel 32473  df-sat 32476
This theorem is referenced by:  satfv1  32496  satfrel  32500  satfdm  32502  satfrnmapom  32503  satfv0fun  32504  satfv0fvfmla0  32546
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