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Theorem satfv0 35001
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 7900 . . . 4 ∅ ∈ ω
2 elelsuc 6447 . . . 4 (∅ ∈ ω → ∅ ∈ suc ω)
31, 2mp1i 13 . . 3 ((𝑀𝑉𝐸𝑊) → ∅ ∈ suc ω)
4 satfv0.s . . . 4 𝑆 = (𝑀 Sat 𝐸)
54satfvsucom 35000 . . 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 1458 . 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 34991 . . . . . . . . . 10 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖𝑔𝑗) ∈ (ω × (ω × ω)))
8 eleq1 2817 . . . . . . . . . 10 (𝑥 = (𝑖𝑔𝑗) → (𝑥 ∈ (ω × (ω × ω)) ↔ (𝑖𝑔𝑗) ∈ (ω × (ω × ω))))
97, 8syl5ibrcom 246 . . . . . . . . 9 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = (𝑖𝑔𝑗) → 𝑥 ∈ (ω × (ω × ω))))
109adantrd 490 . . . . . . . 8 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}) → 𝑥 ∈ (ω × (ω × ω))))
1110pm4.71d 560 . . . . . . 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 3213 . . . . . 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 3222 . . . . . 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 459 . . . . . 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 296 . . . . 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 5219 . . . 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 9674 . . . . 5 ω ∈ V
1817, 17xpex 7761 . . . . 5 (ω × ω) ∈ V
19 xpexg 7758 . . . . . 6 ((ω ∈ V ∧ (ω × ω) ∈ V) → (ω × (ω × ω)) ∈ V)
20 oveq1 7433 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → (𝑖𝑔𝑗) = (𝑚𝑔𝑗))
2120eqeq2d 2739 . . . . . . . . . . . . 13 (𝑖 = 𝑚 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑚𝑔𝑗)))
22 fveq2 6902 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑚 → (𝑎𝑖) = (𝑎𝑚))
2322breq1d 5162 . . . . . . . . . . . . . . 15 (𝑖 = 𝑚 → ((𝑎𝑖)𝐸(𝑎𝑗) ↔ (𝑎𝑚)𝐸(𝑎𝑗)))
2423rabbidv 3438 . . . . . . . . . . . . . 14 (𝑖 = 𝑚 → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)})
2524eqeq2d 2739 . . . . . . . . . . . . 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 7434 . . . . . . . . . . . . . 14 (𝑗 = 𝑛 → (𝑚𝑔𝑗) = (𝑚𝑔𝑛))
2827eqeq2d 2739 . . . . . . . . . . . . 13 (𝑗 = 𝑛 → (𝑥 = (𝑚𝑔𝑗) ↔ 𝑥 = (𝑚𝑔𝑛)))
29 fveq2 6902 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑛 → (𝑎𝑗) = (𝑎𝑛))
3029breq2d 5164 . . . . . . . . . . . . . . 15 (𝑗 = 𝑛 → ((𝑎𝑚)𝐸(𝑎𝑗) ↔ (𝑎𝑚)𝐸(𝑎𝑛)))
3130rabbidv 3438 . . . . . . . . . . . . . 14 (𝑗 = 𝑛 → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑗)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)})
3231eqeq2d 2739 . . . . . . . . . . . . 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, 33cbvrex2vw 3237 . . . . . . . . . . 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 2732 . . . . . . . . . . . . . . . . . . 19 (𝑥 = (𝑖𝑔𝑗) → (𝑥 = (𝑚𝑔𝑛) ↔ (𝑖𝑔𝑗) = (𝑚𝑔𝑛)))
3635adantl 480 . . . . . . . . . . . . . . . . . 18 ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖𝑔𝑗)) → (𝑥 = (𝑚𝑔𝑛) ↔ (𝑖𝑔𝑗) = (𝑚𝑔𝑛)))
37 goeleq12bg 34992 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑚𝑔𝑛) ↔ (𝑖 = 𝑚𝑗 = 𝑛)))
3822eqcomd 2734 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑚 → (𝑎𝑚) = (𝑎𝑖))
3929eqcomd 2734 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = 𝑛 → (𝑎𝑛) = (𝑎𝑗))
4038, 39breqan12d 5168 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖 = 𝑚𝑗 = 𝑛) → ((𝑎𝑚)𝐸(𝑎𝑛) ↔ (𝑎𝑖)𝐸(𝑎𝑗)))
4140rabbidv 3438 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖 = 𝑚𝑗 = 𝑛) → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})
4237, 41biimtrdi 252 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑚𝑔𝑛) → {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑚)𝐸(𝑎𝑛)} = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))
4342imp 405 . . . . . . . . . . . . . . . . . . . . 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 2745 . . . . . . . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . 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 3209 . . . . . . . . . . . . 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 405 . . . . . . . . 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 2732 . . . . . . . . . . 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 3217 . . . . . . . . 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 2555 . . . . . . . 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 5465 . . . . . . 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 7975 . . . . 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 690 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (ω × (ω × ω)) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)}))} ∈ V
6716, 66eqeltri 2825 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ (𝑎𝑖)𝐸(𝑎𝑗)})} ∈ V
6867rdg0 8448 . 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 2784 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 394  wo 845  wal 1531   = wceq 1533  wcel 2098  ∃*wmo 2527  {cab 2705  wral 3058  wrex 3067  {crab 3430  Vcvv 3473  cdif 3946  cun 3947  cin 3948  c0 4326  {csn 4632  cop 4638   class class class wbr 5152  {copab 5214  cmpt 5235   × cxp 5680  cres 5684  suc csuc 6376  cfv 6553  (class class class)co 7426  ωcom 7876  1st c1st 7997  2nd c2nd 7998  reccrdg 8436  m cmap 8851  𝑔cgoe 34976  𝑔cgna 34977  𝑔cgol 34978   Sat csat 34979
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-goel 34983  df-sat 34986
This theorem is referenced by:  satfv1  35006  satfrel  35010  satfdm  35012  satfrnmapom  35013  satfv0fun  35014  satfv0fvfmla0  35056
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