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Theorem satfbrsuc 32498
Description: The binary relation of a satisfaction predicate as function over wff codes at a successor. (Contributed by AV, 13-Oct-2023.)
Hypotheses
Ref Expression
satfbrsuc.s 𝑆 = (𝑀 Sat 𝐸)
satfbrsuc.p 𝑃 = (𝑆𝑁)
Assertion
Ref Expression
satfbrsuc (((𝑀𝑉𝐸𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴𝑋𝐵𝑌)) → (𝐴(𝑆‘suc 𝑁)𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢𝑃 (∃𝑣𝑃 (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))))
Distinct variable groups:   𝐴,𝑖,𝑢,𝑣   𝐵,𝑖,𝑢,𝑣   𝑓,𝐸,𝑖,𝑢,𝑣,𝑧   𝑓,𝑀,𝑖,𝑢,𝑣,𝑧   𝑢,𝑁,𝑣   𝑣,𝑃   𝑢,𝑆,𝑣   𝑢,𝑉   𝑢,𝑊
Allowed substitution hints:   𝐴(𝑧,𝑓)   𝐵(𝑧,𝑓)   𝑃(𝑧,𝑢,𝑓,𝑖)   𝑆(𝑧,𝑓,𝑖)   𝑁(𝑧,𝑓,𝑖)   𝑉(𝑧,𝑣,𝑓,𝑖)   𝑊(𝑧,𝑣,𝑓,𝑖)   𝑋(𝑧,𝑣,𝑢,𝑓,𝑖)   𝑌(𝑧,𝑣,𝑢,𝑓,𝑖)

Proof of Theorem satfbrsuc
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 satfbrsuc.s . . . . . 6 𝑆 = (𝑀 Sat 𝐸)
21satfvsuc 32493 . . . . 5 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → (𝑆‘suc 𝑁) = ((𝑆𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}))
323expa 1112 . . . 4 (((𝑀𝑉𝐸𝑊) ∧ 𝑁 ∈ ω) → (𝑆‘suc 𝑁) = ((𝑆𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}))
433adant3 1126 . . 3 (((𝑀𝑉𝐸𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴𝑋𝐵𝑌)) → (𝑆‘suc 𝑁) = ((𝑆𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}))
54breqd 5073 . 2 (((𝑀𝑉𝐸𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴𝑋𝐵𝑌)) → (𝐴(𝑆‘suc 𝑁)𝐵𝐴((𝑆𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})𝐵))
6 brun 5113 . . . 4 (𝐴((𝑆𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})𝐵 ↔ (𝐴(𝑆𝑁)𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}𝐵))
7 satfbrsuc.p . . . . . . . 8 𝑃 = (𝑆𝑁)
87eqcomi 2833 . . . . . . 7 (𝑆𝑁) = 𝑃
98breqi 5068 . . . . . 6 (𝐴(𝑆𝑁)𝐵𝐴𝑃𝐵)
109a1i 11 . . . . 5 ((𝐴𝑋𝐵𝑌) → (𝐴(𝑆𝑁)𝐵𝐴𝑃𝐵))
11 eqeq1 2828 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝐴 = ((1st𝑢)⊼𝑔(1st𝑣))))
12 eqeq1 2828 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ↔ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
1311, 12bi2anan9 635 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))))
1413rexbidv 3301 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑣𝑃 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ∃𝑣𝑃 (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))))
15 eqeq1 2828 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝐴 = ∀𝑔𝑖(1st𝑢)))
16 eqeq1 2828 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ↔ 𝐵 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))
1715, 16bi2anan9 635 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ (𝐴 = ∀𝑔𝑖(1st𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
1817rexbidv 3301 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}) ↔ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
1914, 18orbi12d 914 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → ((∃𝑣𝑃 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ (∃𝑣𝑃 (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))))
2019rexbidv 3301 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (∃𝑢𝑃 (∃𝑣𝑃 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∃𝑢𝑃 (∃𝑣𝑃 (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))))
218rexeqi 3419 . . . . . . . . 9 (∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ∃𝑣𝑃 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
2221orbi1i 909 . . . . . . . 8 ((∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ (∃𝑣𝑃 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
238, 22rexeqbii 3330 . . . . . . 7 (∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ↔ ∃𝑢𝑃 (∃𝑣𝑃 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))
2423opabbii 5129 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢𝑃 (∃𝑣𝑃 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}
2520, 24brabga 5417 . . . . 5 ((𝐴𝑋𝐵𝑌) → (𝐴{⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}𝐵 ↔ ∃𝑢𝑃 (∃𝑣𝑃 (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))))
2610, 25orbi12d 914 . . . 4 ((𝐴𝑋𝐵𝑌) → ((𝐴(𝑆𝑁)𝐵𝐴{⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}𝐵) ↔ (𝐴𝑃𝐵 ∨ ∃𝑢𝑃 (∃𝑣𝑃 (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))))
276, 26syl5bb 284 . . 3 ((𝐴𝑋𝐵𝑌) → (𝐴((𝑆𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢𝑃 (∃𝑣𝑃 (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))))
28273ad2ant3 1129 . 2 (((𝑀𝑉𝐸𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴𝑋𝐵𝑌)) → (𝐴((𝑆𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆𝑁)(∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))})𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢𝑃 (∃𝑣𝑃 (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))))
295, 28bitrd 280 1 (((𝑀𝑉𝐸𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴𝑋𝐵𝑌)) → (𝐴(𝑆‘suc 𝑁)𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢𝑃 (∃𝑣𝑃 (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀𝑔𝑖(1st𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 843  w3a 1081   = wceq 1530  wcel 2106  wral 3142  wrex 3143  {crab 3146  cdif 3936  cun 3937  cin 3938  {csn 4563  cop 4569   class class class wbr 5062  {copab 5124  cres 5555  suc csuc 6190  cfv 6351  (class class class)co 7151  ωcom 7571  1st c1st 7681  2nd c2nd 7682  m cmap 8399  𝑔cgna 32466  𝑔cgol 32467   Sat csat 32468
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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  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 2615  df-eu 2649  df-clab 2803  df-cleq 2817  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-sat 32475
This theorem is referenced by: (None)
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