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Theorem satfv0 33992
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 7830 . . . 4 ∅ ∈ ω
2 elelsuc 6395 . . . 4 (∅ ∈ ω → ∅ ∈ suc ω)
31, 2mp1i 13 . . 3 ((𝑀𝑉𝐸𝑊) → ∅ ∈ suc ω)
4 satfv0.s . . . 4 𝑆 = (𝑀 Sat 𝐸)
54satfvsucom 33991 . . 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 33982 . . . . . . . . . 10 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) ∈ (ω × (ω × ω)))
8 eleq1 2826 . . . . . . . . . 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 493 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → 𝑥 ∈ (ω × (ω × ω))))
1110pm4.71d 563 . . . . . . 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 3210 . . . . . 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 3218 . . . . . 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 462 . . . . . 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 5177 . . . 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 9586 . . . . 5 ω ∈ V
1817, 17xpex 7692 . . . . 5 (ω × ω) ∈ V
19 xpexg 7689 . . . . . 6 ((ω ∈ V ∧ (ω × ω) ∈ V) → (ω × (ω × ω)) ∈ V)
20 oveq1 7369 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → (𝑖𝑔𝑗) = (𝑚𝑔𝑗))
2120eqeq2d 2748 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑚𝑔𝑗)))
22 fveq2 6847 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑚 → (𝑎𝑖) = (𝑎𝑚))
2322breq1d 5120 . . . . . . . . . . . . . . 15 (𝑖 = 𝑚 → ((𝑎𝑖)𝐸(𝑎𝑗) ↔ (𝑎𝑚)𝐸(𝑎𝑗)))
2423rabbidv 3418 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)})
2524eqeq2d 2748 . . . . . . . . . . . . 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 7370 . . . . . . . . . . . . . 14 (𝑗 = 𝑛 → (𝑚𝑔𝑗) = (𝑚𝑔𝑛))
2827eqeq2d 2748 . . . . . . . . . . . . 13 (𝑗 = 𝑛 → (𝑥 = (𝑚𝑔𝑗) ↔ 𝑥 = (𝑚𝑔𝑛)))
29 fveq2 6847 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑛 → (𝑎𝑗) = (𝑎𝑛))
3029breq2d 5122 . . . . . . . . . . . . . . 15 (𝑗 = 𝑛 → ((𝑎𝑚)𝐸(𝑎𝑗) ↔ (𝑎𝑚)𝐸(𝑎𝑛)))
3130rabbidv 3418 . . . . . . . . . . . . . 14 (𝑗 = 𝑛 → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)})
3231eqeq2d 2748 . . . . . . . . . . . . 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 3231 . . . . . . . . . . 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 2741 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑖𝑔𝑗) → (𝑥 = (𝑚𝑔𝑛) ↔ (𝑖𝑔𝑗) = (𝑚𝑔𝑛)))
3635adantl 483 . . . . . . . . . . . . . . . . . 18 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖𝑔𝑗)) → (𝑥 = (𝑚𝑔𝑛) ↔ (𝑖𝑔𝑗) = (𝑚𝑔𝑛)))
37 goeleq12bg 33983 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑚𝑔𝑛) ↔ (𝑖 = 𝑚𝑗 = 𝑛)))
3822eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑚 → (𝑎𝑚) = (𝑎𝑖))
3929eqcomd 2743 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑛 → (𝑎𝑛) = (𝑎𝑗))
4038, 39breqan12d 5126 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 = 𝑚𝑗 = 𝑛) → ((𝑎𝑚)𝐸(𝑎𝑛) ↔ (𝑎𝑖)𝐸(𝑎𝑗)))
4140rabbidv 3418 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 = 𝑚𝑗 = 𝑛) → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})
4237, 41syl6bi 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 408 . . . . . . . . . . . . . . . . . . . . 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 2754 . . . . . . . . . . . . . . . . . . . . 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 432 . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . 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 239 . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . 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 455 . . . . . . . . . . . . . 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 3206 . . . . . . . . . . . . 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 3197 . . . . . . . . . . 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 216 . . . . . . . . . 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 408 . . . . . . . . 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 1799 . . . . . . . 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 2741 . . . . . . . . . . 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 3214 . . . . . . . . 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 230 . . . . . . 7 ∃*𝑦𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})
63 moabex 5421 . . . . . . 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 7903 . . . . 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 2834 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})} ∈ V
6867rdg0 8372 . 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 2793 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 205  wa 397  wo 846  wal 1540   = wceq 1542  wcel 2107  ∃*wmo 2537  {cab 2714  wral 3065  wrex 3074  {crab 3410  Vcvv 3448  cdif 3912  cun 3913  cin 3914  c0 4287  {csn 4591  cop 4597   class class class wbr 5110  {copab 5172  cmpt 5193   × cxp 5636  cres 5640  suc csuc 6324  cfv 6501  (class class class)co 7362  ωcom 7807  1st c1st 7924  2nd c2nd 7925  reccrdg 8360  m cmap 8772  𝑔cgoe 33967  𝑔cgna 33968  𝑔cgol 33969   Sat csat 33970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-goel 33974  df-sat 33977
This theorem is referenced by:  satfv1  33997  satfrel  34001  satfdm  34003  satfrnmapom  34004  satfv0fun  34005  satfv0fvfmla0  34047
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