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Theorem satfbrsuc 32741
 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 32736 . . . . 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 1115 . . . 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 1129 . . 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 5042 . 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 5082 . . . 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 2807 . . . . . . 7 (𝑆𝑁) = 𝑃
98breqi 5037 . . . . . 6 (𝐴(𝑆𝑁)𝐵𝐴𝑃𝐵)
109a1i 11 . . . . 5 ((𝐴𝑋𝐵𝑌) → (𝐴(𝑆𝑁)𝐵𝐴𝑃𝐵))
11 eqeq1 2802 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝐴 = ((1st𝑢)⊼𝑔(1st𝑣))))
12 eqeq1 2802 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ↔ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
1311, 12bi2anan9 638 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))))
1413rexbidv 3256 . . . . . . . 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 2802 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝐴 = ∀𝑔𝑖(1st𝑢)))
16 eqeq1 2802 . . . . . . . . . 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 638 . . . . . . . . 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 3256 . . . . . . . 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 916 . . . . . . 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 3256 . . . . . 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 3363 . . . . . . . . 9 (∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ∃𝑣𝑃 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
2221orbi1i 911 . . . . . . . 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 3285 . . . . . . 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 5098 . . . . . 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 5387 . . . . 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 916 . . . 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 286 . . 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 1132 . 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 282 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 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880  {csn 4525  ⟨cop 4531   class class class wbr 5031  {copab 5093   ↾ cres 5522  suc csuc 6162  ‘cfv 6325  (class class class)co 7136  ωcom 7563  1st c1st 7672  2nd c2nd 7673   ↑m cmap 8392  ⊼𝑔cgna 32709  ∀𝑔cgol 32710   Sat csat 32711 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-inf2 9091 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-sat 32718 This theorem is referenced by: (None)
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