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Theorem satfrnmapom 32644
Description: The range of the satisfaction predicate as function over wff codes in any model 𝑀 and any binary relation 𝐸 on 𝑀 for a natural number 𝑁 is a subset of the power set of all mappings from the natural numbers into the model 𝑀. (Contributed by AV, 13-Oct-2023.)
Assertion
Ref Expression
satfrnmapom ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → ran ((𝑀 Sat 𝐸)‘𝑁) ⊆ 𝒫 (𝑀m ω))

Proof of Theorem satfrnmapom
Dummy variables 𝑎 𝑏 𝑓 𝑖 𝑗 𝑢 𝑣 𝑥 𝑦 𝑧 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6659 . . . . . . . . 9 (𝑎 = ∅ → ((𝑀 Sat 𝐸)‘𝑎) = ((𝑀 Sat 𝐸)‘∅))
21rneqd 5796 . . . . . . . 8 (𝑎 = ∅ → ran ((𝑀 Sat 𝐸)‘𝑎) = ran ((𝑀 Sat 𝐸)‘∅))
32eleq2d 2901 . . . . . . 7 (𝑎 = ∅ → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑎) ↔ 𝑛 ∈ ran ((𝑀 Sat 𝐸)‘∅)))
43imbi1d 345 . . . . . 6 (𝑎 = ∅ → ((𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑎) → 𝑛 ∈ 𝒫 (𝑀m ω)) ↔ (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘∅) → 𝑛 ∈ 𝒫 (𝑀m ω))))
54imbi2d 344 . . . . 5 (𝑎 = ∅ → (((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑎) → 𝑛 ∈ 𝒫 (𝑀m ω))) ↔ ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘∅) → 𝑛 ∈ 𝒫 (𝑀m ω)))))
6 fveq2 6659 . . . . . . . . 9 (𝑎 = 𝑏 → ((𝑀 Sat 𝐸)‘𝑎) = ((𝑀 Sat 𝐸)‘𝑏))
76rneqd 5796 . . . . . . . 8 (𝑎 = 𝑏 → ran ((𝑀 Sat 𝐸)‘𝑎) = ran ((𝑀 Sat 𝐸)‘𝑏))
87eleq2d 2901 . . . . . . 7 (𝑎 = 𝑏 → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑎) ↔ 𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏)))
98imbi1d 345 . . . . . 6 (𝑎 = 𝑏 → ((𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑎) → 𝑛 ∈ 𝒫 (𝑀m ω)) ↔ (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω))))
109imbi2d 344 . . . . 5 (𝑎 = 𝑏 → (((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑎) → 𝑛 ∈ 𝒫 (𝑀m ω))) ↔ ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω)))))
11 fveq2 6659 . . . . . . . . 9 (𝑎 = suc 𝑏 → ((𝑀 Sat 𝐸)‘𝑎) = ((𝑀 Sat 𝐸)‘suc 𝑏))
1211rneqd 5796 . . . . . . . 8 (𝑎 = suc 𝑏 → ran ((𝑀 Sat 𝐸)‘𝑎) = ran ((𝑀 Sat 𝐸)‘suc 𝑏))
1312eleq2d 2901 . . . . . . 7 (𝑎 = suc 𝑏 → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑎) ↔ 𝑛 ∈ ran ((𝑀 Sat 𝐸)‘suc 𝑏)))
1413imbi1d 345 . . . . . 6 (𝑎 = suc 𝑏 → ((𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑎) → 𝑛 ∈ 𝒫 (𝑀m ω)) ↔ (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘suc 𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω))))
1514imbi2d 344 . . . . 5 (𝑎 = suc 𝑏 → (((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑎) → 𝑛 ∈ 𝒫 (𝑀m ω))) ↔ ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘suc 𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω)))))
16 fveq2 6659 . . . . . . . . 9 (𝑎 = 𝑁 → ((𝑀 Sat 𝐸)‘𝑎) = ((𝑀 Sat 𝐸)‘𝑁))
1716rneqd 5796 . . . . . . . 8 (𝑎 = 𝑁 → ran ((𝑀 Sat 𝐸)‘𝑎) = ran ((𝑀 Sat 𝐸)‘𝑁))
1817eleq2d 2901 . . . . . . 7 (𝑎 = 𝑁 → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑎) ↔ 𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑁)))
1918imbi1d 345 . . . . . 6 (𝑎 = 𝑁 → ((𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑎) → 𝑛 ∈ 𝒫 (𝑀m ω)) ↔ (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑁) → 𝑛 ∈ 𝒫 (𝑀m ω))))
2019imbi2d 344 . . . . 5 (𝑎 = 𝑁 → (((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑎) → 𝑛 ∈ 𝒫 (𝑀m ω))) ↔ ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑁) → 𝑛 ∈ 𝒫 (𝑀m ω)))))
21 eqid 2824 . . . . . . . . 9 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
2221satfv0 32632 . . . . . . . 8 ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})})
2322rneqd 5796 . . . . . . 7 ((𝑀𝑉𝐸𝑊) → ran ((𝑀 Sat 𝐸)‘∅) = ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})})
2423eleq2d 2901 . . . . . 6 ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘∅) ↔ 𝑛 ∈ ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})}))
25 rnopab 5814 . . . . . . . 8 ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})} = {𝑦 ∣ ∃𝑥𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})}
2625eleq2i 2907 . . . . . . 7 (𝑛 ∈ ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})} ↔ 𝑛 ∈ {𝑦 ∣ ∃𝑥𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})})
27 vex 3483 . . . . . . . . . 10 𝑛 ∈ V
28 eqeq1 2828 . . . . . . . . . . . . 13 (𝑦 = 𝑛 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} ↔ 𝑛 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}))
2928anbi2d 631 . . . . . . . . . . . 12 (𝑦 = 𝑛 → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ↔ (𝑥 = (𝑖𝑔𝑗) ∧ 𝑛 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})))
30292rexbidv 3293 . . . . . . . . . . 11 (𝑦 = 𝑛 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑛 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})))
3130exbidv 1923 . . . . . . . . . 10 (𝑦 = 𝑛 → (∃𝑥𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ↔ ∃𝑥𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑛 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})))
3227, 31elab 3653 . . . . . . . . 9 (𝑛 ∈ {𝑦 ∣ ∃𝑥𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})} ↔ ∃𝑥𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑛 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}))
33 ovex 7179 . . . . . . . . . . . . . . 15 (𝑀m ω) ∈ V
34 ssrab2 4042 . . . . . . . . . . . . . . 15 {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} ⊆ (𝑀m ω)
3533, 34elpwi2 5236 . . . . . . . . . . . . . 14 {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} ∈ 𝒫 (𝑀m ω)
36 eleq1 2903 . . . . . . . . . . . . . 14 (𝑛 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} → (𝑛 ∈ 𝒫 (𝑀m ω) ↔ {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} ∈ 𝒫 (𝑀m ω)))
3735, 36mpbiri 261 . . . . . . . . . . . . 13 (𝑛 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} → 𝑛 ∈ 𝒫 (𝑀m ω))
3837adantl 485 . . . . . . . . . . . 12 ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑛 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) → 𝑛 ∈ 𝒫 (𝑀m ω))
3938a1i 11 . . . . . . . . . . 11 ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑛 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) → 𝑛 ∈ 𝒫 (𝑀m ω)))
4039rexlimivv 3285 . . . . . . . . . 10 (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑛 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) → 𝑛 ∈ 𝒫 (𝑀m ω))
4140exlimiv 1932 . . . . . . . . 9 (∃𝑥𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑛 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) → 𝑛 ∈ 𝒫 (𝑀m ω))
4232, 41sylbi 220 . . . . . . . 8 (𝑛 ∈ {𝑦 ∣ ∃𝑥𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})} → 𝑛 ∈ 𝒫 (𝑀m ω))
4342a1i 11 . . . . . . 7 ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ {𝑦 ∣ ∃𝑥𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})} → 𝑛 ∈ 𝒫 (𝑀m ω)))
4426, 43syl5bi 245 . . . . . 6 ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})} → 𝑛 ∈ 𝒫 (𝑀m ω)))
4524, 44sylbid 243 . . . . 5 ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘∅) → 𝑛 ∈ 𝒫 (𝑀m ω)))
4621satfvsuc 32635 . . . . . . . . . . . . . . 15 ((𝑀𝑉𝐸𝑊𝑏 ∈ ω) → ((𝑀 Sat 𝐸)‘suc 𝑏) = (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))}))
47463expa 1115 . . . . . . . . . . . . . 14 (((𝑀𝑉𝐸𝑊) ∧ 𝑏 ∈ ω) → ((𝑀 Sat 𝐸)‘suc 𝑏) = (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))}))
4847rneqd 5796 . . . . . . . . . . . . 13 (((𝑀𝑉𝐸𝑊) ∧ 𝑏 ∈ ω) → ran ((𝑀 Sat 𝐸)‘suc 𝑏) = ran (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))}))
49 rnun 5992 . . . . . . . . . . . . 13 ran (((𝑀 Sat 𝐸)‘𝑏) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))}) = (ran ((𝑀 Sat 𝐸)‘𝑏) ∪ ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))})
5048, 49syl6eq 2875 . . . . . . . . . . . 12 (((𝑀𝑉𝐸𝑊) ∧ 𝑏 ∈ ω) → ran ((𝑀 Sat 𝐸)‘suc 𝑏) = (ran ((𝑀 Sat 𝐸)‘𝑏) ∪ ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))}))
5150eleq2d 2901 . . . . . . . . . . 11 (((𝑀𝑉𝐸𝑊) ∧ 𝑏 ∈ ω) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘suc 𝑏) ↔ 𝑛 ∈ (ran ((𝑀 Sat 𝐸)‘𝑏) ∪ ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))})))
52 elun 4111 . . . . . . . . . . . 12 (𝑛 ∈ (ran ((𝑀 Sat 𝐸)‘𝑏) ∪ ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))}) ↔ (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) ∨ 𝑛 ∈ ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))}))
53 rnopab 5814 . . . . . . . . . . . . . . 15 ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))} = {𝑦 ∣ ∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))}
5453eleq2i 2907 . . . . . . . . . . . . . 14 (𝑛 ∈ ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))} ↔ 𝑛 ∈ {𝑦 ∣ ∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))})
55 eqeq1 2828 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑛 → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ↔ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
5655anbi2d 631 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑛 → ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))))
5756rexbidv 3290 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))))
58 eqeq1 2828 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑛 → (𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ↔ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))
5958anbi2d 631 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑛 → ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
6059rexbidv 3290 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑛 → (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
6157, 60orbi12d 916 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑛 → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)})) ↔ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))))
6261rexbidv 3290 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑛 → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)})) ↔ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))))
6362exbidv 1923 . . . . . . . . . . . . . . 15 (𝑦 = 𝑛 → (∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)})) ↔ ∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))))
6427, 63elab 3653 . . . . . . . . . . . . . 14 (𝑛 ∈ {𝑦 ∣ ∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))} ↔ ∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
6554, 64bitri 278 . . . . . . . . . . . . 13 (𝑛 ∈ ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))} ↔ ∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
6665orbi2i 910 . . . . . . . . . . . 12 ((𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) ∨ 𝑛 ∈ ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))}) ↔ (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) ∨ ∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))))
6752, 66bitri 278 . . . . . . . . . . 11 (𝑛 ∈ (ran ((𝑀 Sat 𝐸)‘𝑏) ∪ ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(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)}))}) ↔ (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) ∨ ∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))))
6851, 67syl6bb 290 . . . . . . . . . 10 (((𝑀𝑉𝐸𝑊) ∧ 𝑏 ∈ ω) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘suc 𝑏) ↔ (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) ∨ ∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))))
6968expcom 417 . . . . . . . . 9 (𝑏 ∈ ω → ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘suc 𝑏) ↔ (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) ∨ ∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))))))
7069adantr 484 . . . . . . . 8 ((𝑏 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω)))) → ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘suc 𝑏) ↔ (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) ∨ ∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))))))
7170imp 410 . . . . . . 7 (((𝑏 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω)))) ∧ (𝑀𝑉𝐸𝑊)) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘suc 𝑏) ↔ (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) ∨ ∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))))
72 simpr 488 . . . . . . . . 9 ((𝑏 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω)))) → ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω))))
7372imp 410 . . . . . . . 8 (((𝑏 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω)))) ∧ (𝑀𝑉𝐸𝑊)) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω)))
74 difss 4094 . . . . . . . . . . . . . . . . . 18 ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ⊆ (𝑀m ω)
7533, 74elpwi2 5236 . . . . . . . . . . . . . . . . 17 ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ 𝒫 (𝑀m ω)
76 eleq1 2903 . . . . . . . . . . . . . . . . 17 (𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) → (𝑛 ∈ 𝒫 (𝑀m ω) ↔ ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ 𝒫 (𝑀m ω)))
7775, 76mpbiri 261 . . . . . . . . . . . . . . . 16 (𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) → 𝑛 ∈ 𝒫 (𝑀m ω))
7877adantl 485 . . . . . . . . . . . . . . 15 ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → 𝑛 ∈ 𝒫 (𝑀m ω))
7978adantl 485 . . . . . . . . . . . . . 14 ((𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏) ∧ (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))) → 𝑛 ∈ 𝒫 (𝑀m ω))
8079rexlimiva 3274 . . . . . . . . . . . . 13 (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) → 𝑛 ∈ 𝒫 (𝑀m ω))
81 ssrab2 4042 . . . . . . . . . . . . . . . . . 18 {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ⊆ (𝑀m ω)
8233, 81elpwi2 5236 . . . . . . . . . . . . . . . . 17 {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∈ 𝒫 (𝑀m ω)
83 eleq1 2903 . . . . . . . . . . . . . . . . 17 (𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → (𝑛 ∈ 𝒫 (𝑀m ω) ↔ {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∈ 𝒫 (𝑀m ω)))
8482, 83mpbiri 261 . . . . . . . . . . . . . . . 16 (𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} → 𝑛 ∈ 𝒫 (𝑀m ω))
8584adantl 485 . . . . . . . . . . . . . . 15 ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) → 𝑛 ∈ 𝒫 (𝑀m ω))
8685a1i 11 . . . . . . . . . . . . . 14 (𝑖 ∈ ω → ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) → 𝑛 ∈ 𝒫 (𝑀m ω)))
8786rexlimiv 3273 . . . . . . . . . . . . 13 (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) → 𝑛 ∈ 𝒫 (𝑀m ω))
8880, 87jaoi 854 . . . . . . . . . . . 12 ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → 𝑛 ∈ 𝒫 (𝑀m ω))
8988a1i 11 . . . . . . . . . . 11 (𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → 𝑛 ∈ 𝒫 (𝑀m ω)))
9089rexlimiv 3273 . . . . . . . . . 10 (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → 𝑛 ∈ 𝒫 (𝑀m ω))
9190exlimiv 1932 . . . . . . . . 9 (∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → 𝑛 ∈ 𝒫 (𝑀m ω))
9291a1i 11 . . . . . . . 8 (((𝑏 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω)))) ∧ (𝑀𝑉𝐸𝑊)) → (∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) → 𝑛 ∈ 𝒫 (𝑀m ω)))
9373, 92jaod 856 . . . . . . 7 (((𝑏 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω)))) ∧ (𝑀𝑉𝐸𝑊)) → ((𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) ∨ ∃𝑥𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑏)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑏)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑛 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑛 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))) → 𝑛 ∈ 𝒫 (𝑀m ω)))
9471, 93sylbid 243 . . . . . 6 (((𝑏 ∈ ω ∧ ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω)))) ∧ (𝑀𝑉𝐸𝑊)) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘suc 𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω)))
9594exp31 423 . . . . 5 (𝑏 ∈ ω → (((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω))) → ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘suc 𝑏) → 𝑛 ∈ 𝒫 (𝑀m ω)))))
965, 10, 15, 20, 45, 95finds 7600 . . . 4 (𝑁 ∈ ω → ((𝑀𝑉𝐸𝑊) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑁) → 𝑛 ∈ 𝒫 (𝑀m ω))))
9796com12 32 . . 3 ((𝑀𝑉𝐸𝑊) → (𝑁 ∈ ω → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑁) → 𝑛 ∈ 𝒫 (𝑀m ω))))
98973impia 1114 . 2 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → (𝑛 ∈ ran ((𝑀 Sat 𝐸)‘𝑁) → 𝑛 ∈ 𝒫 (𝑀m ω)))
9998ssrdv 3959 1 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → ran ((𝑀 Sat 𝐸)‘𝑁) ⊆ 𝒫 (𝑀m ω))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wral 3133  wrex 3134  {crab 3137  Vcvv 3480  cdif 3916  cun 3917  cin 3918  wss 3919  c0 4276  𝒫 cpw 4522  {csn 4550  cop 4556   class class class wbr 5053  {copab 5115  ran crn 5544  cres 5545  suc csuc 6181  cfv 6344  (class class class)co 7146  ωcom 7571  1st c1st 7679  2nd c2nd 7680  m cmap 8398  𝑔cgoe 32607  𝑔cgna 32608  𝑔cgol 32609   Sat csat 32610
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-inf2 9097
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-om 7572  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-goel 32614  df-sat 32617
This theorem is referenced by:  satff  32684
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