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Theorem satfdmlem 34814
Description: Lemma for satfdm 34815. (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 34813 . . . . 5 ((𝑀𝑉𝐸𝑊𝑌 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑌))
21adantr 480 . . . 4 (((𝑀𝑉𝐸𝑊𝑌 ∈ ω) ∧ dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌)) → Rel ((𝑀 Sat 𝐸)‘𝑌))
3 1stdm 8019 . . . 4 ((Rel ((𝑀 Sat 𝐸)‘𝑌) ∧ 𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (1st𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌))
42, 3sylan 579 . . 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 2814 . . . . . 6 (dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌) → ((1st𝑢) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) ↔ (1st𝑢) ∈ dom ((𝑁 Sat 𝐹)‘𝑌)))
65adantl 481 . . . . 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 480 . . . 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 6894 . . . . . 6 (1st𝑢) ∈ V
9 eldm2g 5889 . . . . . 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 484 . . . . . . . 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 729 . . . . . . . . . . . 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 8019 . . . . . . . . . . . 12 ((Rel ((𝑀 Sat 𝐸)‘𝑌) ∧ 𝑣 ∈ ((𝑀 Sat 𝐸)‘𝑌)) → (1st𝑣) ∈ dom ((𝑀 Sat 𝐸)‘𝑌))
1412, 13sylancom 587 . . . . . . . . . . 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 2814 . . . . . . . . . . . . 13 (dom ((𝑀 Sat 𝐸)‘𝑌) = dom ((𝑁 Sat 𝐹)‘𝑌) → ((1st𝑣) ∈ dom ((𝑀 Sat 𝐸)‘𝑌) ↔ (1st𝑣) ∈ dom ((𝑁 Sat 𝐹)‘𝑌)))
1615ad5antlr 732 . . . . . . . . . . . 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 6894 . . . . . . . . . . . . . 14 (1st𝑣) ∈ V
18 eldm2g 5889 . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . 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 3470 . . . . . . . . . . . . . . . . . . . . . 22 𝑠 ∈ V
228, 21op1std 7978 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = ⟨(1st𝑢), 𝑠⟩ → (1st𝑎) = (1st𝑢))
2322eqcomd 2730 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = ⟨(1st𝑢), 𝑠⟩ → (1st𝑢) = (1st𝑎))
2423ad3antlr 728 . . . . . . . . . . . . . . . . . . 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 3470 . . . . . . . . . . . . . . . . . . . . 21 𝑟 ∈ V
2617, 25op1std 7978 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = ⟨(1st𝑣), 𝑟⟩ → (1st𝑏) = (1st𝑣))
2726eqcomd 2730 . . . . . . . . . . . . . . . . . . 19 (𝑏 = ⟨(1st𝑣), 𝑟⟩ → (1st𝑣) = (1st𝑏))
2824, 27oveqan12d 7420 . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 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 3595 . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . 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 1928 . . . . . . . . . . . . 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 3147 . . . . . . . . 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 2725 . . . . . . . . . . . . . 14 (𝑎 = ⟨(1st𝑢), 𝑠⟩ → 𝑖 = 𝑖)
3938, 23goaleq12d 34797 . . . . . . . . . . . . 13 (𝑎 = ⟨(1st𝑢), 𝑠⟩ → ∀𝑔𝑖(1st𝑢) = ∀𝑔𝑖(1st𝑎))
4039eqeq2d 2735 . . . . . . . . . . . 12 (𝑎 = ⟨(1st𝑢), 𝑠⟩ → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝑥 = ∀𝑔𝑖(1st𝑎)))
4140biimpd 228 . . . . . . . . . . 11 (𝑎 = ⟨(1st𝑢), 𝑠⟩ → (𝑥 = ∀𝑔𝑖(1st𝑢) → 𝑥 = ∀𝑔𝑖(1st𝑎)))
4241adantl 481 . . . . . . . . . 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 3162 . . . . . . . . 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 961 . . . . . . . 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 3595 . . . . . . 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 412 . . . . . 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 1928 . . . . 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 3147 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 395  wo 844  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wrex 3062  Vcvv 3466  cop 4626  dom cdm 5666  Rel wrel 5671  cfv 6533  (class class class)co 7401  ωcom 7848  1st c1st 7966  𝑔cgna 34780  𝑔cgol 34781   Sat csat 34782
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 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9631
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 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-goel 34786  df-goal 34788  df-sat 34789
This theorem is referenced by:  satfdm  34815
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