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Theorem satffunlem1lem2 32829
 Description: Lemma 2 for satffunlem1 32833. (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 7594 . . . 4 ∅ ∈ ω
2 satfdmfmla 32826 . . . 4 ((𝑀𝑉𝐸𝑊 ∧ ∅ ∈ ω) → dom ((𝑀 Sat 𝐸)‘∅) = (Fmla‘∅))
31, 2mp3an3 1447 . . 3 ((𝑀𝑉𝐸𝑊) → dom ((𝑀 Sat 𝐸)‘∅) = (Fmla‘∅))
4 ovex 7178 . . . . . . . . . 10 (𝑀m ω) ∈ V
54difexi 5200 . . . . . . . . 9 ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ V
65a1i 11 . . . . . . . 8 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ V)
76ralrimiva 3149 . . . . . . 7 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → ∀𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ V)
84rabex 5203 . . . . . . . . 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 3149 . . . . . . 7 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → ∀𝑖 ∈ ω {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} ∈ V)
117, 10jca 515 . . . . . 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 3149 . . . . 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 5756 . . . . 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 32793 . . . . . . . . . . . . 13 ((𝑀𝑉𝐸𝑊 ∧ ∅ ∈ ω) → Rel ((𝑀 Sat 𝐸)‘∅))
161, 15mp3an3 1447 . . . . . . . . . . . 12 ((𝑀𝑉𝐸𝑊) → Rel ((𝑀 Sat 𝐸)‘∅))
17 1stdm 7734 . . . . . . . . . . . 12 ((Rel ((𝑀 Sat 𝐸)‘∅) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st𝑢) ∈ dom ((𝑀 Sat 𝐸)‘∅))
1816, 17sylan 583 . . . . . . . . . . 11 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st𝑢) ∈ dom ((𝑀 Sat 𝐸)‘∅))
192eqcomd 2804 . . . . . . . . . . . . 13 ((𝑀𝑉𝐸𝑊 ∧ ∅ ∈ ω) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
201, 19mp3an3 1447 . . . . . . . . . . . 12 ((𝑀𝑉𝐸𝑊) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
2120adantr 484 . . . . . . . . . . 11 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
2218, 21eleqtrrd 2893 . . . . . . . . . 10 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st𝑢) ∈ (Fmla‘∅))
2322adantr 484 . . . . . . . . 9 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))) → (1st𝑢) ∈ (Fmla‘∅))
24 oveq1 7152 . . . . . . . . . . . . 13 (𝑓 = (1st𝑢) → (𝑓𝑔𝑔) = ((1st𝑢)⊼𝑔𝑔))
2524eqeq2d 2809 . . . . . . . . . . . 12 (𝑓 = (1st𝑢) → (𝑥 = (𝑓𝑔𝑔) ↔ 𝑥 = ((1st𝑢)⊼𝑔𝑔)))
2625rexbidv 3257 . . . . . . . . . . 11 (𝑓 = (1st𝑢) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ↔ ∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔)))
27 eqidd 2799 . . . . . . . . . . . . . 14 (𝑓 = (1st𝑢) → 𝑖 = 𝑖)
28 id 22 . . . . . . . . . . . . . 14 (𝑓 = (1st𝑢) → 𝑓 = (1st𝑢))
2927, 28goaleq12d 32777 . . . . . . . . . . . . 13 (𝑓 = (1st𝑢) → ∀𝑔𝑖𝑓 = ∀𝑔𝑖(1st𝑢))
3029eqeq2d 2809 . . . . . . . . . . . 12 (𝑓 = (1st𝑢) → (𝑥 = ∀𝑔𝑖𝑓𝑥 = ∀𝑔𝑖(1st𝑢)))
3130rexbidv 3257 . . . . . . . . . . 11 (𝑓 = (1st𝑢) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓 ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))
3226, 31orbi12d 916 . . . . . . . . . 10 (𝑓 = (1st𝑢) → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ↔ (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
3332adantl 485 . . . . . . . . 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 7734 . . . . . . . . . . . . . . . . 17 ((Rel ((𝑀 Sat 𝐸)‘∅) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st𝑣) ∈ dom ((𝑀 Sat 𝐸)‘∅))
3516, 34sylan 583 . . . . . . . . . . . . . . . 16 (((𝑀𝑉𝐸𝑊) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (1st𝑣) ∈ dom ((𝑀 Sat 𝐸)‘∅))
3620adantr 484 . . . . . . . . . . . . . . . 16 (((𝑀𝑉𝐸𝑊) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
3735, 36eleqtrrd 2893 . . . . . . . . . . . . . . 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 7153 . . . . . . . . . . . . . . . 16 (𝑔 = (1st𝑣) → ((1st𝑢)⊼𝑔𝑔) = ((1st𝑢)⊼𝑔(1st𝑣)))
4039eqeq2d 2809 . . . . . . . . . . . . . . 15 (𝑔 = (1st𝑣) → (𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
4140adantl 485 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . 14 (((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))
4338, 41, 42rspcedvd 3575 . . . . . . . . . . . . 13 (((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → ∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔))
4443ex 416 . . . . . . . . . . . 12 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔)))
4544rexlimdva 3244 . . . . . . . . . . 11 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔)))
4645orim1d 963 . . . . . . . . . 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 410 . . . . . . . . 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 3575 . . . . . . . 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 416 . . . . . . 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 3244 . . . . . 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 7737 . . . . . . . . . 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 2875 . . . . . . . . 9 ((𝑀𝑉𝐸𝑊) → (𝑓 ∈ dom ((𝑀 Sat 𝐸)‘∅) ↔ 𝑓 ∈ (Fmla‘∅)))
5452, 53bitr3d 284 . . . . . . . 8 ((𝑀𝑉𝐸𝑊) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑢) = 𝑓𝑓 ∈ (Fmla‘∅)))
55 r19.41v 3301 . . . . . . . . . 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 7152 . . . . . . . . . . . . . . . . 17 ((1st𝑢) = 𝑓 → ((1st𝑢)⊼𝑔𝑔) = (𝑓𝑔𝑔))
5756eqeq2d 2809 . . . . . . . . . . . . . . . 16 ((1st𝑢) = 𝑓 → (𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ 𝑥 = (𝑓𝑔𝑔)))
5857rexbidv 3257 . . . . . . . . . . . . . . 15 ((1st𝑢) = 𝑓 → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ ∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔)))
59 eqidd 2799 . . . . . . . . . . . . . . . . . 18 ((1st𝑢) = 𝑓𝑖 = 𝑖)
60 id 22 . . . . . . . . . . . . . . . . . 18 ((1st𝑢) = 𝑓 → (1st𝑢) = 𝑓)
6159, 60goaleq12d 32777 . . . . . . . . . . . . . . . . 17 ((1st𝑢) = 𝑓 → ∀𝑔𝑖(1st𝑢) = ∀𝑔𝑖𝑓)
6261eqeq2d 2809 . . . . . . . . . . . . . . . 16 ((1st𝑢) = 𝑓 → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝑥 = ∀𝑔𝑖𝑓))
6362rexbidv 3257 . . . . . . . . . . . . . . 15 ((1st𝑢) = 𝑓 → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓))
6458, 63orbi12d 916 . . . . . . . . . . . . . 14 ((1st𝑢) = 𝑓 → ((∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ (∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
6564adantl 485 . . . . . . . . . . . . 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 2804 . . . . . . . . . . . . . . . . . . . 20 ((𝑀𝑉𝐸𝑊) → (Fmla‘∅) = dom ((𝑀 Sat 𝐸)‘∅))
6766eleq2d 2875 . . . . . . . . . . . . . . . . . . 19 ((𝑀𝑉𝐸𝑊) → (𝑔 ∈ (Fmla‘∅) ↔ 𝑔 ∈ dom ((𝑀 Sat 𝐸)‘∅)))
68 releldm2 7737 . . . . . . . . . . . . . . . . . . . 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 282 . . . . . . . . . . . . . . . . . 18 ((𝑀𝑉𝐸𝑊) → (𝑔 ∈ (Fmla‘∅) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑣) = 𝑔))
71 r19.41v 3301 . . . . . . . . . . . . . . . . . . . 20 (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) ↔ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)))
7239eqcoms 2806 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1st𝑣) = 𝑔 → ((1st𝑢)⊼𝑔𝑔) = ((1st𝑢)⊼𝑔(1st𝑣)))
7372eqeq2d 2809 . . . . . . . . . . . . . . . . . . . . . . 23 ((1st𝑣) = 𝑔 → (𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
7473biimpa 480 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) → 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))
7574a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝑀𝑉𝐸𝑊) → (((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) → 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
7675reximdv 3233 . . . . . . . . . . . . . . . . . . . 20 ((𝑀𝑉𝐸𝑊) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
7771, 76syl5bir 246 . . . . . . . . . . . . . . . . . . 19 ((𝑀𝑉𝐸𝑊) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
7877expd 419 . . . . . . . . . . . . . . . . . 18 ((𝑀𝑉𝐸𝑊) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(1st𝑣) = 𝑔 → (𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))))
7970, 78sylbid 243 . . . . . . . . . . . . . . . . 17 ((𝑀𝑉𝐸𝑊) → (𝑔 ∈ (Fmla‘∅) → (𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))))
8079rexlimdv 3243 . . . . . . . . . . . . . . . 16 ((𝑀𝑉𝐸𝑊) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
8180adantr 484 . . . . . . . . . . . . . . 15 (((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
8281adantr 484 . . . . . . . . . . . . . 14 ((((𝑀𝑉𝐸𝑊) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)) ∧ (1st𝑢) = 𝑓) → (∃𝑔 ∈ (Fmla‘∅)𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
8382orim1d 963 . . . . . . . . . . . . 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 263 . . . . . . . . . . . 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 457 . . . . . . . . . . 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 3234 . . . . . . . . . 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, 86syl5bir 246 . . . . . . . . 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 419 . . . . . . . 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 263 . . . . . . 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 3243 . . . . . 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 215 . . . . 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 2862 . . . 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 2833 . . 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 4143 . 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 32824 . 2 ((Fmla‘∅) ∩ {𝑥 ∣ ∃𝑓 ∈ (Fmla‘∅)(∃𝑔 ∈ (Fmla‘∅)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)}) = ∅
9694, 95eqtrdi 2849 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 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3442   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882  ∅c0 4246  {csn 4528  ⟨cop 4534  {copab 5096  dom cdm 5523   ↾ cres 5525  Rel wrel 5528  ‘cfv 6332  (class class class)co 7145  ωcom 7573  1st c1st 7682  2nd c2nd 7683   ↑m cmap 8407  ⊼𝑔cgna 32760  ∀𝑔cgol 32761   Sat csat 32762  Fmlacfmla 32763 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 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-inf2 9106 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 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-2o 8104  df-map 8409  df-goel 32766  df-gona 32767  df-goal 32768  df-sat 32769  df-fmla 32771 This theorem is referenced by:  satffunlem1  32833
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