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Theorem satfdmlem 34296
Description: Lemma for satfdm 34297. (Contributed by AV, 12-Oct-2023.)
Assertion
Ref Expression
satfdmlem (((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎))))
Distinct variable groups:   𝐸,𝑎,𝑏,𝑖,𝑢,𝑣   𝐹,𝑎,𝑏,𝑖,𝑢,𝑣   𝑀,𝑎,𝑏,𝑖,𝑢,𝑣   𝑁,𝑎,𝑏,𝑖,𝑢,𝑣   𝑉,𝑎,𝑏,𝑖,𝑢,𝑣   𝑊,𝑎,𝑏,𝑖,𝑢,𝑣   𝑌,𝑎,𝑏,𝑖,𝑢,𝑣   𝑥,𝑎,𝑏,𝑖,𝑢,𝑣
Allowed substitution hints:   𝐸(𝑥)   𝐹(𝑥)   𝑀(𝑥)   𝑁(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑌(𝑥)

Proof of Theorem satfdmlem
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 satfrel 34295 . . . . 5 ((𝑀𝑉𝐸𝑊𝑌 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑌))
21adantr 482 . . . 4 (((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → Rel ((𝑀 Sat 𝐸)‘𝑌))
3 1stdm 8020 . . . 4 ((Rel ((𝑀 Sat 𝐸)‘𝑌) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (1st𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌))
42, 3sylan 581 . . 3 ((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (1st𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌))
5 eleq2 2823 . . . . . 6 (dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌) → ((1st𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) ↔ (1st𝑢) ∈ dom ((𝑁 Sat 𝐹)‘𝑌)))
65adantl 483 . . . . 5 (((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → ((1st𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) ↔ (1st𝑢) ∈ dom ((𝑁 Sat 𝐹)‘𝑌)))
76adantr 482 . . . 4 ((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((1st𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) ↔ (1st𝑢) ∈ dom ((𝑁 Sat 𝐹)‘𝑌)))
8 fvex 6900 . . . . . 6 (1st𝑢) ∈ V
9 eldm2g 5896 . . . . . 6 ((1st𝑢) ∈ V → ((1st𝑢) ∈ dom ((𝑁 Sat 𝐹)‘𝑌) ↔ ∃𝑠⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)))
108, 9ax-mp 5 . . . . 5 ((1st𝑢) ∈ dom ((𝑁 Sat 𝐹)‘𝑌) ↔ ∃𝑠⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌))
11 simpr 486 . . . . . . . 8 (((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) → ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌))
122ad4antr 731 . . . . . . . . . . . 12 (((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → Rel ((𝑀 Sat 𝐸)‘𝑌))
13 1stdm 8020 . . . . . . . . . . . 12 ((Rel ((𝑀 Sat 𝐸)‘𝑌) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (1st𝑣) ∈ dom ((𝑀 Sat 𝐸)‘𝑌))
1412, 13sylancom 589 . . . . . . . . . . 11 (((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (1st𝑣) ∈ dom ((𝑀 Sat 𝐸)‘𝑌))
15 eleq2 2823 . . . . . . . . . . . . 13 (dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌) → ((1st𝑣) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) ↔ (1st𝑣) ∈ dom ((𝑁 Sat 𝐹)‘𝑌)))
1615ad5antlr 734 . . . . . . . . . . . 12 (((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((1st𝑣) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) ↔ (1st𝑣) ∈ dom ((𝑁 Sat 𝐹)‘𝑌)))
17 fvex 6900 . . . . . . . . . . . . . 14 (1st𝑣) ∈ V
18 eldm2g 5896 . . . . . . . . . . . . . 14 ((1st𝑣) ∈ V → ((1st𝑣) ∈ dom ((𝑁 Sat 𝐹)‘𝑌) ↔ ∃𝑟⟨(1st𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)))
1917, 18ax-mp 5 . . . . . . . . . . . . 13 ((1st𝑣) ∈ dom ((𝑁 Sat 𝐹)‘𝑌) ↔ ∃𝑟⟨(1st𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌))
20 simpr 486 . . . . . . . . . . . . . . . 16 ((((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) → ⟨(1st𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌))
21 vex 3479 . . . . . . . . . . . . . . . . . . . . . 22 𝑠 ∈ V
228, 21op1std 7979 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = ⟨(1st𝑢), 𝑠⟩ → (1st𝑎) = (1st𝑢))
2322eqcomd 2739 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = ⟨(1st𝑢), 𝑠⟩ → (1st𝑢) = (1st𝑎))
2423ad3antlr 730 . . . . . . . . . . . . . . . . . . 19 ((((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) → (1st𝑢) = (1st𝑎))
25 vex 3479 . . . . . . . . . . . . . . . . . . . . 21 𝑟 ∈ V
2617, 25op1std 7979 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = ⟨(1st𝑣), 𝑟⟩ → (1st𝑏) = (1st𝑣))
2726eqcomd 2739 . . . . . . . . . . . . . . . . . . 19 (𝑏 = ⟨(1st𝑣), 𝑟⟩ → (1st𝑣) = (1st𝑏))
2824, 27oveqan12d 7422 . . . . . . . . . . . . . . . . . 18 (((((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑏 = ⟨(1st𝑣), 𝑟⟩) → ((1st𝑢)⊼𝑔(1st𝑣)) = ((1st𝑎)⊼𝑔(1st𝑏)))
2928eqeq2d 2744 . . . . . . . . . . . . . . . . 17 (((((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑏 = ⟨(1st𝑣), 𝑟⟩) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ↔ 𝑥 = ((1st𝑎)⊼𝑔(1st𝑏))))
3029biimpd 228 . . . . . . . . . . . . . . . 16 (((((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑏 = ⟨(1st𝑣), 𝑟⟩) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → 𝑥 = ((1st𝑎)⊼𝑔(1st𝑏))))
3120, 30rspcimedv 3602 . . . . . . . . . . . . . . 15 ((((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏))))
3231ex 414 . . . . . . . . . . . . . 14 (((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (⟨(1st𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)))))
3332exlimdv 1937 . . . . . . . . . . . . 13 (((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (∃𝑟⟨(1st𝑣), 𝑟⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)))))
3419, 33biimtrid 241 . . . . . . . . . . . 12 (((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((1st𝑣) ∈ dom ((𝑁 Sat 𝐹)‘𝑌) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)))))
3516, 34sylbid 239 . . . . . . . . . . 11 (((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((1st𝑣) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)))))
3614, 35mpd 15 . . . . . . . . . 10 (((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏))))
3736rexlimdva 3156 . . . . . . . . 9 ((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏))))
38 eqidd 2734 . . . . . . . . . . . . . 14 (𝑎 = ⟨(1st𝑢), 𝑠⟩ → 𝑖 = 𝑖)
3938, 23goaleq12d 34279 . . . . . . . . . . . . 13 (𝑎 = ⟨(1st𝑢), 𝑠⟩ → ∀𝑔𝑖(1st𝑢) = ∀𝑔𝑖(1st𝑎))
4039eqeq2d 2744 . . . . . . . . . . . 12 (𝑎 = ⟨(1st𝑢), 𝑠⟩ → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝑥 = ∀𝑔𝑖(1st𝑎)))
4140biimpd 228 . . . . . . . . . . 11 (𝑎 = ⟨(1st𝑢), 𝑠⟩ → (𝑥 = ∀𝑔𝑖(1st𝑢) → 𝑥 = ∀𝑔𝑖(1st𝑎)))
4241adantl 483 . . . . . . . . . 10 ((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) → (𝑥 = ∀𝑔𝑖(1st𝑢) → 𝑥 = ∀𝑔𝑖(1st𝑎)))
4342reximdv 3171 . . . . . . . . 9 ((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢) → ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎)))
4437, 43orim12d 964 . . . . . . . 8 ((((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑎 = ⟨(1st𝑢), 𝑠⟩) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → (∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎))))
4511, 44rspcimedv 3602 . . . . . . 7 (((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) ∧ ⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌)) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎))))
4645ex 414 . . . . . 6 ((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎)))))
4746exlimdv 1937 . . . . 5 ((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (∃𝑠⟨(1st𝑢), 𝑠⟩ ∈ ((𝑁 Sat 𝐹)‘𝑌) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎)))))
4810, 47biimtrid 241 . . . 4 ((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((1st𝑢) ∈ dom ((𝑁 Sat 𝐹)‘𝑌) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎)))))
497, 48sylbid 239 . . 3 ((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((1st𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎)))))
504, 49mpd 15 . 2 ((((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎))))
5150rexlimdva 3156 1 (((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑎 ∈ ((𝑁 Sat 𝐹)‘𝑌)(∃𝑏 ∈ ((𝑁 Sat 𝐹)‘𝑌)𝑥 = ((1st𝑎)⊼𝑔(1st𝑏)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑎))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wex 1782  wcel 2107  wrex 3071  Vcvv 3475  cop 4632  dom cdm 5674  Rel wrel 5679  cfv 6539  (class class class)co 7403  ωcom 7849  1st c1st 7967  𝑔cgna 34262  𝑔cgol 34263   Sat csat 34264
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 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719  ax-inf2 9631
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-int 4949  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6296  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8260  df-wrecs 8291  df-recs 8365  df-rdg 8404  df-goel 34268  df-goal 34270  df-sat 34271
This theorem is referenced by:  satfdm  34297
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