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Theorem satfbrsuc 34848
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 34843 . . . . 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 5150 . 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 5190 . . . 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 2733 . . . . . . 7 (𝑆𝑁) = 𝑃
98breqi 5145 . . . . . 6 (𝐴(𝑆𝑁)𝐵𝐴𝑃𝐵)
109a1i 11 . . . . 5 ((𝐴𝑋𝐵𝑌) → (𝐴(𝑆𝑁)𝐵𝐴𝑃𝐵))
11 eqeq1 2728 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝐴 = ((1st𝑢)⊼𝑔(1st𝑣))))
12 eqeq1 2728 . . . . . . . . . 10 (𝑦 = 𝐵 → (𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ↔ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
1311, 12bi2anan9 636 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ (𝐴 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝐵 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))))))
1413rexbidv 3170 . . . . . . . 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 2728 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝐴 = ∀𝑔𝑖(1st𝑢)))
16 eqeq1 2728 . . . . . . . . . 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 636 . . . . . . . . 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 3170 . . . . . . . 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 915 . . . . . . 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 3170 . . . . . 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 3316 . . . . . . . . 9 (∃𝑣 ∈ (𝑆𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ↔ ∃𝑣𝑃 (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))
2221orbi1i 910 . . . . . . . 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 3331 . . . . . . 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 5206 . . . . . 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 5525 . . . . 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 915 . . . 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, 26bitrid 283 . . 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 279 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 205  wa 395  wo 844  w3a 1084   = wceq 1533  wcel 2098  wral 3053  wrex 3062  {crab 3424  cdif 3938  cun 3939  cin 3940  {csn 4621  cop 4627   class class class wbr 5139  {copab 5201  cres 5669  suc csuc 6357  cfv 6534  (class class class)co 7402  ωcom 7849  1st c1st 7967  2nd c2nd 7968  m cmap 8817  𝑔cgna 34816  𝑔cgol 34817   Sat csat 34818
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 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-inf2 9633
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-sat 34825
This theorem is referenced by: (None)
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