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Theorem satffunlem1lem2 33801
Description: Lemma 2 for satffunlem1 33805. (Contributed by AV, 23-Oct-2023.)
Assertion
Ref Expression
satffunlem1lem2 ((𝑀𝑉𝐸𝑊) → (dom ((𝑀 Sat 𝐸)‘∅) ∩ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}) = ∅)
Distinct variable groups:   𝑓,𝐸,𝑖,𝑢,𝑣,𝑥,𝑦   𝑓,𝑀,𝑖,𝑢,𝑣,𝑥,𝑦   𝑓,𝑉,𝑖,𝑢,𝑣,𝑥   𝑓,𝑊,𝑖,𝑢,𝑣,𝑥   𝑥,𝑗,𝑦
Allowed substitution hints:   𝐸(𝑗)   𝑀(𝑗)   𝑉(𝑦,𝑗)   𝑊(𝑦,𝑗)

Proof of Theorem satffunlem1lem2
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 peano1 7818 . . . 4 ∅ ∈ ω
2 satfdmfmla 33798 . . . 4 ((𝑀𝑉𝐸𝑊 ∧ ∅ ∈ ω) → dom ((𝑀 Sat 𝐸)‘∅) = (Fmla‘∅))
31, 2mp3an3 1451 . . 3 ((𝑀𝑉𝐸𝑊) → dom ((𝑀 Sat 𝐸)‘∅) = (Fmla‘∅))
4 ovex 7385 . . . . . . . . . 10 (𝑀m ω) ∈ V
54difexi 5284 . . . . . . . . 9 ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ V
65a1i 11 . . . . . . . 8 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ V)
76ralrimiva 3142 . . . . . . 7 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → ∀𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ V)
84rabex 5288 . . . . . . . . 9 {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∈ V
98a1i 11 . . . . . . . 8 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑖 ∈ ω) → {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∈ V)
109ralrimiva 3142 . . . . . . 7 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → ∀𝑖 ∈ ω {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∈ V)
117, 10jca 513 . . . . . 6 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (∀𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ V ∧ ∀𝑖 ∈ ω {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∈ V))
1211ralrimiva 3142 . . . . 5 ((𝑀𝑉𝐸𝑊) → ∀𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∀𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ V ∧ ∀𝑖 ∈ ω {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∈ V))
13 dmopab2rex 5872 . . . . 5 (∀𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∀𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ V ∧ ∀𝑖 ∈ ω {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∈ V) → dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))} = {𝑥 ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))})
1412, 13syl 17 . . . 4 ((𝑀𝑉𝐸𝑊) → dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))} = {𝑥 ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))})
15 satfrel 33765 . . . . . . . . . . . . 13 ((𝑀𝑉𝐸𝑊 ∧ ∅ ∈ ω) → Rel ((𝑀 Sat 𝐸)‘∅))
161, 15mp3an3 1451 . . . . . . . . . . . 12 ((𝑀𝑉𝐸𝑊) → Rel ((𝑀 Sat 𝐸)‘∅))
17 1stdm 7965 . . . . . . . . . . . 12 ((Rel ((𝑀 Sat 𝐸)‘∅) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st𝑢) ∈ dom ((𝑀 Sat 𝐸)‘∅))
1816, 17sylan 581 . . . . . . . . . . 11 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st𝑢) ∈ dom ((𝑀 Sat 𝐸)‘∅))
192eqcomd 2744 . . . . . . . . . . . . 13 ((𝑀𝑉𝐸𝑊 ∧ ∅ ∈ ω) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
201, 19mp3an3 1451 . . . . . . . . . . . 12 ((𝑀𝑉𝐸𝑊) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
2120adantr 482 . . . . . . . . . . 11 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
2218, 21eleqtrrd 2842 . . . . . . . . . 10 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st𝑢) ∈ (Fmla‘∅))
2322adantr 482 . . . . . . . . 9 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))) → (1st𝑢) ∈ (Fmla‘∅))
24 oveq1 7359 . . . . . . . . . . . . 13 (𝑓 = (1st𝑢) → (𝑓𝑔𝑔) = ((1st𝑢)⊼𝑔𝑔))
2524eqeq2d 2749 . . . . . . . . . . . 12 (𝑓 = (1st𝑢) → (𝑥 = (𝑓𝑔𝑔) ↔ 𝑥 = ((1st𝑢)⊼𝑔𝑔)))
2625rexbidv 3174 . . . . . . . . . . 11 (𝑓 = (1st𝑢) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ↔ ∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔)))
27 eqidd 2739 . . . . . . . . . . . . . 14 (𝑓 = (1st𝑢) → 𝑖 = 𝑖)
28 id 22 . . . . . . . . . . . . . 14 (𝑓 = (1st𝑢) → 𝑓 = (1st𝑢))
2927, 28goaleq12d 33749 . . . . . . . . . . . . 13 (𝑓 = (1st𝑢) → ∀𝑔𝑖𝑓 = ∀𝑔𝑖(1st𝑢))
3029eqeq2d 2749 . . . . . . . . . . . 12 (𝑓 = (1st𝑢) → (𝑥 = ∀𝑔𝑖𝑓𝑥 = ∀𝑔𝑖(1st𝑢)))
3130rexbidv 3174 . . . . . . . . . . 11 (𝑓 = (1st𝑢) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓 ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))
3226, 31orbi12d 918 . . . . . . . . . 10 (𝑓 = (1st𝑢) → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ↔ (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
3332adantl 483 . . . . . . . . 9 (((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))) ∧ 𝑓 = (1st𝑢)) → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ↔ (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
34 1stdm 7965 . . . . . . . . . . . . . . . . 17 ((Rel ((𝑀 Sat 𝐸)‘∅) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st𝑣) ∈ dom ((𝑀 Sat 𝐸)‘∅))
3516, 34sylan 581 . . . . . . . . . . . . . . . 16 (((𝑀𝑉𝐸𝑊) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st𝑣) ∈ dom ((𝑀 Sat 𝐸)‘∅))
3620adantr 482 . . . . . . . . . . . . . . . 16 (((𝑀𝑉𝐸𝑊) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
3735, 36eleqtrrd 2842 . . . . . . . . . . . . . . 15 (((𝑀𝑉𝐸𝑊) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st𝑣) ∈ (Fmla‘∅))
3837ad4ant13 750 . . . . . . . . . . . . . 14 (((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → (1st𝑣) ∈ (Fmla‘∅))
39 oveq2 7360 . . . . . . . . . . . . . . . 16 (𝑔 = (1st𝑣) → ((1st𝑢)⊼𝑔𝑔) = ((1st𝑢)⊼𝑔(1st𝑣)))
4039eqeq2d 2749 . . . . . . . . . . . . . . 15 (𝑔 = (1st𝑣) → (𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
4140adantl 483 . . . . . . . . . . . . . 14 ((((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) ∧ 𝑔 = (1st𝑣)) → (𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
42 simpr 486 . . . . . . . . . . . . . 14 (((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))
4338, 41, 42rspcedvd 3582 . . . . . . . . . . . . 13 (((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → ∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔))
4443ex 414 . . . . . . . . . . . 12 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔)))
4544rexlimdva 3151 . . . . . . . . . . 11 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔)))
4645orim1d 965 . . . . . . . . . 10 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
4746imp 408 . . . . . . . . 9 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))
4823, 33, 47rspcedvd 3582 . . . . . . . 8 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))) → ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓))
4948ex 414 . . . . . . 7 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
5049rexlimdva 3151 . . . . . 6 ((𝑀𝑉𝐸𝑊) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
51 releldm2 7968 . . . . . . . . . 10 (Rel ((𝑀 Sat 𝐸)‘∅) → (𝑓 ∈ dom ((𝑀 Sat 𝐸)‘∅) ↔ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑢) = 𝑓))
5216, 51syl 17 . . . . . . . . 9 ((𝑀𝑉𝐸𝑊) → (𝑓 ∈ dom ((𝑀 Sat 𝐸)‘∅) ↔ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑢) = 𝑓))
533eleq2d 2824 . . . . . . . . 9 ((𝑀𝑉𝐸𝑊) → (𝑓 ∈ dom ((𝑀 Sat 𝐸)‘∅) ↔ 𝑓 ∈ (Fmla‘∅)))
5452, 53bitr3d 281 . . . . . . . 8 ((𝑀𝑉𝐸𝑊) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑢) = 𝑓𝑓 ∈ (Fmla‘∅)))
55 r19.41v 3184 . . . . . . . . . 10 (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)((1st𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)) ↔ (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
56 oveq1 7359 . . . . . . . . . . . . . . . . 17 ((1st𝑢) = 𝑓 → ((1st𝑢)⊼𝑔𝑔) = (𝑓𝑔𝑔))
5756eqeq2d 2749 . . . . . . . . . . . . . . . 16 ((1st𝑢) = 𝑓 → (𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ 𝑥 = (𝑓𝑔𝑔)))
5857rexbidv 3174 . . . . . . . . . . . . . . 15 ((1st𝑢) = 𝑓 → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ ∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔)))
59 eqidd 2739 . . . . . . . . . . . . . . . . . 18 ((1st𝑢) = 𝑓𝑖 = 𝑖)
60 id 22 . . . . . . . . . . . . . . . . . 18 ((1st𝑢) = 𝑓 → (1st𝑢) = 𝑓)
6159, 60goaleq12d 33749 . . . . . . . . . . . . . . . . 17 ((1st𝑢) = 𝑓 → ∀𝑔𝑖(1st𝑢) = ∀𝑔𝑖𝑓)
6261eqeq2d 2749 . . . . . . . . . . . . . . . 16 ((1st𝑢) = 𝑓 → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝑥 = ∀𝑔𝑖𝑓))
6362rexbidv 3174 . . . . . . . . . . . . . . 15 ((1st𝑢) = 𝑓 → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓))
6458, 63orbi12d 918 . . . . . . . . . . . . . 14 ((1st𝑢) = 𝑓 → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
6564adantl 483 . . . . . . . . . . . . 13 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (1st𝑢) = 𝑓) → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
663eqcomd 2744 . . . . . . . . . . . . . . . . . . . 20 ((𝑀𝑉𝐸𝑊) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
6766eleq2d 2824 . . . . . . . . . . . . . . . . . . 19 ((𝑀𝑉𝐸𝑊) → (𝑔 ∈ (Fmla‘∅) ↔ 𝑔 ∈ dom ((𝑀 Sat 𝐸)‘∅)))
68 releldm2 7968 . . . . . . . . . . . . . . . . . . . 20 (Rel ((𝑀 Sat 𝐸)‘∅) → (𝑔 ∈ dom ((𝑀 Sat 𝐸)‘∅) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑣) = 𝑔))
6916, 68syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑀𝑉𝐸𝑊) → (𝑔 ∈ dom ((𝑀 Sat 𝐸)‘∅) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑣) = 𝑔))
7067, 69bitrd 279 . . . . . . . . . . . . . . . . . 18 ((𝑀𝑉𝐸𝑊) → (𝑔 ∈ (Fmla‘∅) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑣) = 𝑔))
71 r19.41v 3184 . . . . . . . . . . . . . . . . . . . 20 (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) ↔ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)))
7239eqcoms 2746 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑣) = 𝑔 → ((1st𝑢)⊼𝑔𝑔) = ((1st𝑢)⊼𝑔(1st𝑣)))
7372eqeq2d 2749 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑣) = 𝑔 → (𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
7473biimpa 478 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) → 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))
7574a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀𝑉𝐸𝑊) → (((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) → 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
7675reximdv 3166 . . . . . . . . . . . . . . . . . . . 20 ((𝑀𝑉𝐸𝑊) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
7771, 76biimtrrid 242 . . . . . . . . . . . . . . . . . . 19 ((𝑀𝑉𝐸𝑊) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
7877expd 417 . . . . . . . . . . . . . . . . . 18 ((𝑀𝑉𝐸𝑊) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑣) = 𝑔 → (𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))))
7970, 78sylbid 239 . . . . . . . . . . . . . . . . 17 ((𝑀𝑉𝐸𝑊) → (𝑔 ∈ (Fmla‘∅) → (𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))))
8079rexlimdv 3149 . . . . . . . . . . . . . . . 16 ((𝑀𝑉𝐸𝑊) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
8180adantr 482 . . . . . . . . . . . . . . 15 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
8281adantr 482 . . . . . . . . . . . . . 14 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (1st𝑢) = 𝑓) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
8382orim1d 965 . . . . . . . . . . . . 13 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (1st𝑢) = 𝑓) → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
8465, 83sylbird 260 . . . . . . . . . . . 12 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (1st𝑢) = 𝑓) → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
8584expimpd 455 . . . . . . . . . . 11 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (((1st𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
8685reximdva 3164 . . . . . . . . . 10 ((𝑀𝑉𝐸𝑊) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)((1st𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
8755, 86biimtrrid 242 . . . . . . . . 9 ((𝑀𝑉𝐸𝑊) → ((∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
8887expd 417 . . . . . . . 8 ((𝑀𝑉𝐸𝑊) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑢) = 𝑓 → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))))
8954, 88sylbird 260 . . . . . . 7 ((𝑀𝑉𝐸𝑊) → (𝑓 ∈ (Fmla‘∅) → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))))
9089rexlimdv 3149 . . . . . 6 ((𝑀𝑉𝐸𝑊) → (∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) → ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
9150, 90impbid 211 . . . . 5 ((𝑀𝑉𝐸𝑊) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
9291abbidv 2807 . . . 4 ((𝑀𝑉𝐸𝑊) → {𝑥 ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))} = {𝑥 ∣ ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)})
9314, 92eqtrd 2778 . . 3 ((𝑀𝑉𝐸𝑊) → dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))} = {𝑥 ∣ ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)})
943, 93ineq12d 4172 . 2 ((𝑀𝑉𝐸𝑊) → (dom ((𝑀 Sat 𝐸)‘∅) ∩ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}) = ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)}))
95 fmla0disjsuc 33796 . 2 ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)}) = ∅
9694, 95eqtrdi 2794 1 ((𝑀𝑉𝐸𝑊) → (dom ((𝑀 Sat 𝐸)‘∅) ∩ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}))}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  {cab 2715  wral 3063  wrex 3072  {crab 3406  Vcvv 3444  cdif 3906  cun 3907  cin 3908  c0 4281  {csn 4585  cop 4591  {copab 5166  dom cdm 5632  cres 5634  Rel wrel 5637  cfv 6494  (class class class)co 7352  ωcom 7795  1st c1st 7912  2nd c2nd 7913  m cmap 8724  𝑔cgna 33732  𝑔cgol 33733   Sat csat 33734  Fmlacfmla 33735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665  ax-inf2 9536
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5530  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-we 5589  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6252  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7796  df-1st 7914  df-2nd 7915  df-frecs 8205  df-wrecs 8236  df-recs 8310  df-rdg 8349  df-1o 8405  df-2o 8406  df-map 8726  df-goel 33738  df-gona 33739  df-goal 33740  df-sat 33741  df-fmla 33743
This theorem is referenced by:  satffunlem1  33805
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