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Theorem satffunlem2lem2 32767
Description: Lemma 2 for satffunlem2 32769. (Contributed by AV, 27-Oct-2023.)
Hypotheses
Ref Expression
satffunlem2lem2.s 𝑆 = (𝑀 Sat 𝐸)
satffunlem2lem2.a 𝐴 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))
satffunlem2lem2.b 𝐵 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}
Assertion
Ref Expression
satffunlem2lem2 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (dom (𝑆‘suc 𝑁) ∩ dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴))}) = ∅)
Distinct variable groups:   𝐴,𝑖,𝑥,𝑦   𝑥,𝐵,𝑦   𝑖,𝐸,𝑢,𝑣,𝑥   𝑀,𝑎   𝑖,𝑀,𝑢,𝑣,𝑥   𝑖,𝑁,𝑢,𝑣,𝑥,𝑦   𝑆,𝑖,𝑢,𝑣,𝑥,𝑦   𝑖,𝑉,𝑢,𝑣,𝑥   𝑖,𝑊,𝑢,𝑣,𝑥
Allowed substitution hints:   𝐴(𝑧,𝑣,𝑢,𝑎)   𝐵(𝑧,𝑣,𝑢,𝑖,𝑎)   𝑆(𝑧,𝑎)   𝐸(𝑦,𝑧,𝑎)   𝑀(𝑦,𝑧)   𝑁(𝑧,𝑎)   𝑉(𝑦,𝑧,𝑎)   𝑊(𝑦,𝑧,𝑎)

Proof of Theorem satffunlem2lem2
Dummy variables 𝑓 𝑔 𝑡 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 satffunlem2lem2.s . . . . . 6 𝑆 = (𝑀 Sat 𝐸)
21fveq1i 6650 . . . . 5 (𝑆‘suc 𝑁) = ((𝑀 Sat 𝐸)‘suc 𝑁)
32dmeqi 5741 . . . 4 dom (𝑆‘suc 𝑁) = dom ((𝑀 Sat 𝐸)‘suc 𝑁)
4 simprl 770 . . . . . . 7 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → 𝑀𝑉)
5 simprr 772 . . . . . . 7 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → 𝐸𝑊)
6 peano2 7586 . . . . . . . 8 (𝑁 ∈ ω → suc 𝑁 ∈ ω)
76adantr 484 . . . . . . 7 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → suc 𝑁 ∈ ω)
84, 5, 73jca 1125 . . . . . 6 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (𝑀𝑉𝐸𝑊 ∧ suc 𝑁 ∈ ω))
9 satfdmfmla 32761 . . . . . 6 ((𝑀𝑉𝐸𝑊 ∧ suc 𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘suc 𝑁) = (Fmla‘suc 𝑁))
108, 9syl 17 . . . . 5 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → dom ((𝑀 Sat 𝐸)‘suc 𝑁) = (Fmla‘suc 𝑁))
1110adantr 484 . . . 4 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → dom ((𝑀 Sat 𝐸)‘suc 𝑁) = (Fmla‘suc 𝑁))
123, 11syl5eq 2848 . . 3 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → dom (𝑆‘suc 𝑁) = (Fmla‘suc 𝑁))
13 satffunlem2lem2.a . . . . . . . . . 10 𝐴 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))
14 ovex 7172 . . . . . . . . . . 11 (𝑀m ω) ∈ V
1514difexi 5199 . . . . . . . . . 10 ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) ∈ V
1613, 15eqeltri 2889 . . . . . . . . 9 𝐴 ∈ V
1716a1i 11 . . . . . . . 8 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆‘suc 𝑁)) ∧ 𝑣 ∈ (𝑆‘suc 𝑁)) → 𝐴 ∈ V)
1817ralrimiva 3152 . . . . . . 7 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆‘suc 𝑁)) → ∀𝑣 ∈ (𝑆‘suc 𝑁)𝐴 ∈ V)
19 satffunlem2lem2.b . . . . . . . . . 10 𝐵 = {𝑎 ∈ (𝑀m ω) ∣ ∀𝑧𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}
2019, 14rabex2 5204 . . . . . . . . 9 𝐵 ∈ V
2120a1i 11 . . . . . . . 8 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆‘suc 𝑁)) ∧ 𝑖 ∈ ω) → 𝐵 ∈ V)
2221ralrimiva 3152 . . . . . . 7 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆‘suc 𝑁)) → ∀𝑖 ∈ ω 𝐵 ∈ V)
2318, 22jca 515 . . . . . 6 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆‘suc 𝑁)) → (∀𝑣 ∈ (𝑆‘suc 𝑁)𝐴 ∈ V ∧ ∀𝑖 ∈ ω 𝐵 ∈ V))
2423ralrimiva 3152 . . . . 5 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ∀𝑢 ∈ (𝑆‘suc 𝑁)(∀𝑣 ∈ (𝑆‘suc 𝑁)𝐴 ∈ V ∧ ∀𝑖 ∈ ω 𝐵 ∈ V))
25 simplr 768 . . . . . . 7 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑀𝑉𝐸𝑊))
266ancri 553 . . . . . . . 8 (𝑁 ∈ ω → (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω))
2726ad2antrr 725 . . . . . . 7 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω))
2825, 27jca 515 . . . . . 6 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((𝑀𝑉𝐸𝑊) ∧ (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω)))
29 sssucid 6240 . . . . . 6 𝑁 ⊆ suc 𝑁
301satfsschain 32725 . . . . . 6 (((𝑀𝑉𝐸𝑊) ∧ (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω)) → (𝑁 ⊆ suc 𝑁 → (𝑆𝑁) ⊆ (𝑆‘suc 𝑁)))
3128, 29, 30mpisyl 21 . . . . 5 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑆𝑁) ⊆ (𝑆‘suc 𝑁))
32 dmopab3rexdif 32766 . . . . 5 ((∀𝑢 ∈ (𝑆‘suc 𝑁)(∀𝑣 ∈ (𝑆‘suc 𝑁)𝐴 ∈ V ∧ ∀𝑖 ∈ ω 𝐵 ∈ V) ∧ (𝑆𝑁) ⊆ (𝑆‘suc 𝑁)) → dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴))} = {𝑥 ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))})
3324, 31, 32syl2anc 587 . . . 4 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴))} = {𝑥 ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))})
34 simpr 488 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) → 𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁)))
35 fveqeq2 6658 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑢 → ((1st𝑤) = (1st𝑢) ↔ (1st𝑢) = (1st𝑢)))
3635adantl 485 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) ∧ 𝑤 = 𝑢) → ((1st𝑤) = (1st𝑢) ↔ (1st𝑢) = (1st𝑢)))
37 eqidd 2802 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) → (1st𝑢) = (1st𝑢))
3834, 36, 37rspcedvd 3577 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) → ∃𝑤 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑤) = (1st𝑢))
392funeqi 6349 . . . . . . . . . . . . . . . . . . 19 (Fun (𝑆‘suc 𝑁) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑁))
4039biimpi 219 . . . . . . . . . . . . . . . . . 18 (Fun (𝑆‘suc 𝑁) → Fun ((𝑀 Sat 𝐸)‘suc 𝑁))
4140adantl 485 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → Fun ((𝑀 Sat 𝐸)‘suc 𝑁))
421fveq1i 6650 . . . . . . . . . . . . . . . . . 18 (𝑆𝑁) = ((𝑀 Sat 𝐸)‘𝑁)
4331, 42, 23sstr3g 3962 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁))
4441, 43jca 515 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) ∧ ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)))
4544adantr 484 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) ∧ ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)))
46 funeldmdif 7733 . . . . . . . . . . . . . . 15 ((Fun ((𝑀 Sat 𝐸)‘suc 𝑁) ∧ ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)) → ((1st𝑢) ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁)) ↔ ∃𝑤 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑤) = (1st𝑢)))
4745, 46syl 17 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) → ((1st𝑢) ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁)) ↔ ∃𝑤 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑤) = (1st𝑢)))
4838, 47mpbird 260 . . . . . . . . . . . . 13 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) → (1st𝑢) ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁)))
4948ex 416 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁)) → (1st𝑢) ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁))))
502, 42difeq12i 4051 . . . . . . . . . . . . . 14 ((𝑆‘suc 𝑁) ∖ (𝑆𝑁)) = (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))
5150eleq2i 2884 . . . . . . . . . . . . 13 (𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁)) ↔ 𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁)))
5251a1i 11 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁)) ↔ 𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))))
5311eqcomd 2807 . . . . . . . . . . . . . 14 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (Fmla‘suc 𝑁) = dom ((𝑀 Sat 𝐸)‘suc 𝑁))
54 simpl 486 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → 𝑁 ∈ ω)
554, 5, 543jca 1125 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (𝑀𝑉𝐸𝑊𝑁 ∈ ω))
56 satfdmfmla 32761 . . . . . . . . . . . . . . . . 17 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))
5755, 56syl 17 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))
5857eqcomd 2807 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (Fmla‘𝑁) = dom ((𝑀 Sat 𝐸)‘𝑁))
5958adantr 484 . . . . . . . . . . . . . 14 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (Fmla‘𝑁) = dom ((𝑀 Sat 𝐸)‘𝑁))
6053, 59difeq12d 4054 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)) = (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁)))
6160eleq2d 2878 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((1st𝑢) ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)) ↔ (1st𝑢) ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁))))
6249, 52, 613imtr4d 297 . . . . . . . . . . 11 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁)) → (1st𝑢) ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))))
6362imp 410 . . . . . . . . . 10 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) → (1st𝑢) ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)))
6463adantr 484 . . . . . . . . 9 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ (∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))) → (1st𝑢) ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)))
65 oveq1 7146 . . . . . . . . . . . . 13 (𝑓 = (1st𝑢) → (𝑓𝑔𝑔) = ((1st𝑢)⊼𝑔𝑔))
6665eqeq2d 2812 . . . . . . . . . . . 12 (𝑓 = (1st𝑢) → (𝑥 = (𝑓𝑔𝑔) ↔ 𝑥 = ((1st𝑢)⊼𝑔𝑔)))
6766rexbidv 3259 . . . . . . . . . . 11 (𝑓 = (1st𝑢) → (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ↔ ∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔𝑔)))
68 eqidd 2802 . . . . . . . . . . . . . 14 (𝑓 = (1st𝑢) → 𝑖 = 𝑖)
69 id 22 . . . . . . . . . . . . . 14 (𝑓 = (1st𝑢) → 𝑓 = (1st𝑢))
7068, 69goaleq12d 32712 . . . . . . . . . . . . 13 (𝑓 = (1st𝑢) → ∀𝑔𝑖𝑓 = ∀𝑔𝑖(1st𝑢))
7170eqeq2d 2812 . . . . . . . . . . . 12 (𝑓 = (1st𝑢) → (𝑥 = ∀𝑔𝑖𝑓𝑥 = ∀𝑔𝑖(1st𝑢)))
7271rexbidv 3259 . . . . . . . . . . 11 (𝑓 = (1st𝑢) → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓 ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))
7367, 72orbi12d 916 . . . . . . . . . 10 (𝑓 = (1st𝑢) → ((∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ↔ (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
7473adantl 485 . . . . . . . . 9 ((((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ (∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))) ∧ 𝑓 = (1st𝑢)) → ((∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ↔ (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
754adantr 484 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → 𝑀𝑉)
765adantr 484 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → 𝐸𝑊)
776ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → suc 𝑁 ∈ ω)
7875, 76, 773jca 1125 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑀𝑉𝐸𝑊 ∧ suc 𝑁 ∈ ω))
79 satfrel 32728 . . . . . . . . . . . . . . . . . 18 ((𝑀𝑉𝐸𝑊 ∧ suc 𝑁 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘suc 𝑁))
8078, 79syl 17 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → Rel ((𝑀 Sat 𝐸)‘suc 𝑁))
812releqi 5620 . . . . . . . . . . . . . . . . 17 (Rel (𝑆‘suc 𝑁) ↔ Rel ((𝑀 Sat 𝐸)‘suc 𝑁))
8280, 81sylibr 237 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → Rel (𝑆‘suc 𝑁))
83 1stdm 7725 . . . . . . . . . . . . . . . 16 ((Rel (𝑆‘suc 𝑁) ∧ 𝑣 ∈ (𝑆‘suc 𝑁)) → (1st𝑣) ∈ dom (𝑆‘suc 𝑁))
8482, 83sylan 583 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑣 ∈ (𝑆‘suc 𝑁)) → (1st𝑣) ∈ dom (𝑆‘suc 𝑁))
8512eqcomd 2807 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (Fmla‘suc 𝑁) = dom (𝑆‘suc 𝑁))
8685adantr 484 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑣 ∈ (𝑆‘suc 𝑁)) → (Fmla‘suc 𝑁) = dom (𝑆‘suc 𝑁))
8784, 86eleqtrrd 2896 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑣 ∈ (𝑆‘suc 𝑁)) → (1st𝑣) ∈ (Fmla‘suc 𝑁))
8887ad4ant13 750 . . . . . . . . . . . . 13 ((((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ 𝑣 ∈ (𝑆‘suc 𝑁)) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → (1st𝑣) ∈ (Fmla‘suc 𝑁))
89 oveq2 7147 . . . . . . . . . . . . . . 15 (𝑔 = (1st𝑣) → ((1st𝑢)⊼𝑔𝑔) = ((1st𝑢)⊼𝑔(1st𝑣)))
9089eqeq2d 2812 . . . . . . . . . . . . . 14 (𝑔 = (1st𝑣) → (𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
9190adantl 485 . . . . . . . . . . . . 13 (((((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ 𝑣 ∈ (𝑆‘suc 𝑁)) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) ∧ 𝑔 = (1st𝑣)) → (𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
92 simpr 488 . . . . . . . . . . . . 13 ((((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ 𝑣 ∈ (𝑆‘suc 𝑁)) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))
9388, 91, 92rspcedvd 3577 . . . . . . . . . . . 12 ((((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ 𝑣 ∈ (𝑆‘suc 𝑁)) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → ∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔𝑔))
9493rexlimdva2 3249 . . . . . . . . . . 11 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) → (∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔𝑔)))
9594orim1d 963 . . . . . . . . . 10 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) → ((∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
9695imp 410 . . . . . . . . 9 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ (∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))) → (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))
9764, 74, 96rspcedvd 3577 . . . . . . . 8 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ (∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))) → ∃𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓))
9897rexlimdva2 3249 . . . . . . 7 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → ∃𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
9955adantr 484 . . . . . . . . . . . . . 14 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑀𝑉𝐸𝑊𝑁 ∈ ω))
100 satfrel 32728 . . . . . . . . . . . . . 14 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Rel ((𝑀 Sat 𝐸)‘𝑁))
10199, 100syl 17 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → Rel ((𝑀 Sat 𝐸)‘𝑁))
10242releqi 5620 . . . . . . . . . . . . 13 (Rel (𝑆𝑁) ↔ Rel ((𝑀 Sat 𝐸)‘𝑁))
103101, 102sylibr 237 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → Rel (𝑆𝑁))
104 1stdm 7725 . . . . . . . . . . . 12 ((Rel (𝑆𝑁) ∧ 𝑢 ∈ (𝑆𝑁)) → (1st𝑢) ∈ dom (𝑆𝑁))
105103, 104sylan 583 . . . . . . . . . . 11 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆𝑁)) → (1st𝑢) ∈ dom (𝑆𝑁))
10642dmeqi 5741 . . . . . . . . . . . . . 14 dom (𝑆𝑁) = dom ((𝑀 Sat 𝐸)‘𝑁)
10799, 56syl 17 . . . . . . . . . . . . . 14 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))
108106, 107syl5eq 2848 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → dom (𝑆𝑁) = (Fmla‘𝑁))
109108eqcomd 2807 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (Fmla‘𝑁) = dom (𝑆𝑁))
110109adantr 484 . . . . . . . . . . 11 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆𝑁)) → (Fmla‘𝑁) = dom (𝑆𝑁))
111105, 110eleqtrrd 2896 . . . . . . . . . 10 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆𝑁)) → (1st𝑢) ∈ (Fmla‘𝑁))
112111adantr 484 . . . . . . . . 9 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆𝑁)) ∧ ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → (1st𝑢) ∈ (Fmla‘𝑁))
11366rexbidv 3259 . . . . . . . . . 10 (𝑓 = (1st𝑢) → (∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔) ↔ ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = ((1st𝑢)⊼𝑔𝑔)))
114113adantl 485 . . . . . . . . 9 ((((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆𝑁)) ∧ ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) ∧ 𝑓 = (1st𝑢)) → (∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔) ↔ ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = ((1st𝑢)⊼𝑔𝑔)))
115 simpr 488 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) → 𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁)))
116 fveqeq2 6658 . . . . . . . . . . . . . . . . . . 19 (𝑡 = 𝑣 → ((1st𝑡) = (1st𝑣) ↔ (1st𝑣) = (1st𝑣)))
117116adantl 485 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) ∧ 𝑡 = 𝑣) → ((1st𝑡) = (1st𝑣) ↔ (1st𝑣) = (1st𝑣)))
118 eqidd 2802 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) → (1st𝑣) = (1st𝑣))
119115, 117, 118rspcedvd 3577 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) → ∃𝑡 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑡) = (1st𝑣))
12044adantr 484 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) ∧ ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)))
121 funeldmdif 7733 . . . . . . . . . . . . . . . . . 18 ((Fun ((𝑀 Sat 𝐸)‘suc 𝑁) ∧ ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)) → ((1st𝑣) ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁)) ↔ ∃𝑡 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑡) = (1st𝑣)))
122120, 121syl 17 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) → ((1st𝑣) ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁)) ↔ ∃𝑡 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑡) = (1st𝑣)))
123119, 122mpbird 260 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))) → (1st𝑣) ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁)))
124123ex 416 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁)) → (1st𝑣) ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁))))
12550eleq2i 2884 . . . . . . . . . . . . . . . 16 (𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁)) ↔ 𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁)))
126125a1i 11 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁)) ↔ 𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))))
12710eqcomd 2807 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (Fmla‘suc 𝑁) = dom ((𝑀 Sat 𝐸)‘suc 𝑁))
128127, 58difeq12d 4054 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)) = (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁)))
129128eleq2d 2878 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → ((1st𝑣) ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)) ↔ (1st𝑣) ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁))))
130129adantr 484 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((1st𝑣) ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)) ↔ (1st𝑣) ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁))))
131124, 126, 1303imtr4d 297 . . . . . . . . . . . . . 14 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁)) → (1st𝑣) ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))))
132131adantr 484 . . . . . . . . . . . . 13 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆𝑁)) → (𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁)) → (1st𝑣) ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))))
133132imp 410 . . . . . . . . . . . 12 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆𝑁)) ∧ 𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) → (1st𝑣) ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)))
134133adantr 484 . . . . . . . . . . 11 ((((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆𝑁)) ∧ 𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → (1st𝑣) ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)))
13590adantl 485 . . . . . . . . . . 11 (((((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆𝑁)) ∧ 𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) ∧ 𝑔 = (1st𝑣)) → (𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
136 simpr 488 . . . . . . . . . . 11 ((((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆𝑁)) ∧ 𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))
137134, 135, 136rspcedvd 3577 . . . . . . . . . 10 ((((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆𝑁)) ∧ 𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = ((1st𝑢)⊼𝑔𝑔))
138137r19.29an 3250 . . . . . . . . 9 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆𝑁)) ∧ ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = ((1st𝑢)⊼𝑔𝑔))
139112, 114, 138rspcedvd 3577 . . . . . . . 8 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ (𝑆𝑁)) ∧ ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → ∃𝑓 ∈ (Fmla‘𝑁)∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔))
140139rexlimdva2 3249 . . . . . . 7 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) → ∃𝑓 ∈ (Fmla‘𝑁)∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔)))
14198, 140orim12d 962 . . . . . 6 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) → (∃𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ∨ ∃𝑓 ∈ (Fmla‘𝑁)∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔))))
1428adantr 484 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑀𝑉𝐸𝑊 ∧ suc 𝑁 ∈ ω))
1439eqcomd 2807 . . . . . . . . . . . . 13 ((𝑀𝑉𝐸𝑊 ∧ suc 𝑁 ∈ ω) → (Fmla‘suc 𝑁) = dom ((𝑀 Sat 𝐸)‘suc 𝑁))
144142, 143syl 17 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (Fmla‘suc 𝑁) = dom ((𝑀 Sat 𝐸)‘suc 𝑁))
145107eqcomd 2807 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (Fmla‘𝑁) = dom ((𝑀 Sat 𝐸)‘𝑁))
146144, 145difeq12d 4054 . . . . . . . . . . 11 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)) = (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁)))
147146eleq2d 2878 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)) ↔ 𝑓 ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁))))
148 eqid 2801 . . . . . . . . . . . . 13 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
149148satfsschain 32725 . . . . . . . . . . . 12 (((𝑀𝑉𝐸𝑊) ∧ (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω)) → (𝑁 ⊆ suc 𝑁 → ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)))
15028, 29, 149mpisyl 21 . . . . . . . . . . 11 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁))
151 releldmdifi 7730 . . . . . . . . . . 11 ((Rel ((𝑀 Sat 𝐸)‘suc 𝑁) ∧ ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)) → (𝑓 ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁)) → ∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑢) = 𝑓))
15280, 150, 151syl2anc 587 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑓 ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁)) → ∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑢) = 𝑓))
153147, 152sylbid 243 . . . . . . . . 9 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)) → ∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑢) = 𝑓))
15450eqcomi 2810 . . . . . . . . . . 11 (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁)) = ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))
155154rexeqi 3366 . . . . . . . . . 10 (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑢) = 𝑓 ↔ ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(1st𝑢) = 𝑓)
156 r19.41v 3303 . . . . . . . . . . . 12 (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))((1st𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)) ↔ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(1st𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
157 oveq1 7146 . . . . . . . . . . . . . . . . . . 19 ((1st𝑢) = 𝑓 → ((1st𝑢)⊼𝑔𝑔) = (𝑓𝑔𝑔))
158157eqeq2d 2812 . . . . . . . . . . . . . . . . . 18 ((1st𝑢) = 𝑓 → (𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ 𝑥 = (𝑓𝑔𝑔)))
159158rexbidv 3259 . . . . . . . . . . . . . . . . 17 ((1st𝑢) = 𝑓 → (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ ∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔)))
160 eqidd 2802 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑢) = 𝑓𝑖 = 𝑖)
161 id 22 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑢) = 𝑓 → (1st𝑢) = 𝑓)
162160, 161goaleq12d 32712 . . . . . . . . . . . . . . . . . . 19 ((1st𝑢) = 𝑓 → ∀𝑔𝑖(1st𝑢) = ∀𝑔𝑖𝑓)
163162eqeq2d 2812 . . . . . . . . . . . . . . . . . 18 ((1st𝑢) = 𝑓 → (𝑥 = ∀𝑔𝑖(1st𝑢) ↔ 𝑥 = ∀𝑔𝑖𝑓))
164163rexbidv 3259 . . . . . . . . . . . . . . . . 17 ((1st𝑢) = 𝑓 → (∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢) ↔ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓))
165159, 164orbi12d 916 . . . . . . . . . . . . . . . 16 ((1st𝑢) = 𝑓 → ((∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
166165adantl 485 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ (1st𝑢) = 𝑓) → ((∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ↔ (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)))
167142, 9syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → dom ((𝑀 Sat 𝐸)‘suc 𝑁) = (Fmla‘suc 𝑁))
168167eqcomd 2807 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (Fmla‘suc 𝑁) = dom ((𝑀 Sat 𝐸)‘suc 𝑁))
169168eleq2d 2878 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑔 ∈ (Fmla‘suc 𝑁) ↔ 𝑔 ∈ dom ((𝑀 Sat 𝐸)‘suc 𝑁)))
170 releldm2 7728 . . . . . . . . . . . . . . . . . . . . 21 (Rel ((𝑀 Sat 𝐸)‘suc 𝑁) → (𝑔 ∈ dom ((𝑀 Sat 𝐸)‘suc 𝑁) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(1st𝑣) = 𝑔))
17180, 170syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑔 ∈ dom ((𝑀 Sat 𝐸)‘suc 𝑁) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(1st𝑣) = 𝑔))
172169, 171bitrd 282 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑔 ∈ (Fmla‘suc 𝑁) ↔ ∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(1st𝑣) = 𝑔))
173 r19.41v 3303 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) ↔ (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)))
1741eqcomi 2810 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑀 Sat 𝐸) = 𝑆
175174fveq1i 6650 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 Sat 𝐸)‘suc 𝑁) = (𝑆‘suc 𝑁)
176175rexeqi 3366 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) ↔ ∃𝑣 ∈ (𝑆‘suc 𝑁)((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)))
17789eqcoms 2809 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1st𝑣) = 𝑔 → ((1st𝑢)⊼𝑔𝑔) = ((1st𝑢)⊼𝑔(1st𝑣)))
178177eqeq2d 2812 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1st𝑣) = 𝑔 → (𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
179178biimpa 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) → 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))
180179a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) → 𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
181180reximdv 3235 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑣 ∈ (𝑆‘suc 𝑁)((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) → ∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
182176, 181syl5bi 245 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) → ∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
183173, 182syl5bir 246 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔𝑔)) → ∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
184183expd 419 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(1st𝑣) = 𝑔 → (𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))))
185172, 184sylbid 243 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑔 ∈ (Fmla‘suc 𝑁) → (𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))))
186185rexlimdv 3245 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
187186ad2antrr 725 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ (1st𝑢) = 𝑓) → (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
188187orim1d 963 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ (1st𝑢) = 𝑓) → ((∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) → (∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
189166, 188sylbird 263 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) ∧ (1st𝑢) = 𝑓) → ((∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) → (∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
190189expimpd 457 . . . . . . . . . . . . 13 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))) → (((1st𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)) → (∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
191190reximdva 3236 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))((1st𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)) → ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
192156, 191syl5bir 246 . . . . . . . . . . 11 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(1st𝑢) = 𝑓 ∧ (∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓)) → ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
193192expd 419 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(1st𝑢) = 𝑓 → ((∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) → ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))))
194155, 193syl5bi 245 . . . . . . . . 9 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑢) = 𝑓 → ((∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) → ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))))
195153, 194syld 47 . . . . . . . 8 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)) → ((∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) → ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)))))
196195rexlimdv 3245 . . . . . . 7 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) → ∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢))))
197145eleq2d 2878 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑓 ∈ (Fmla‘𝑁) ↔ 𝑓 ∈ dom ((𝑀 Sat 𝐸)‘𝑁)))
19855, 100syl 17 . . . . . . . . . . . 12 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → Rel ((𝑀 Sat 𝐸)‘𝑁))
199198adantr 484 . . . . . . . . . . 11 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → Rel ((𝑀 Sat 𝐸)‘𝑁))
200 releldm2 7728 . . . . . . . . . . 11 (Rel ((𝑀 Sat 𝐸)‘𝑁) → (𝑓 ∈ dom ((𝑀 Sat 𝐸)‘𝑁) ↔ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(1st𝑢) = 𝑓))
201199, 200syl 17 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑓 ∈ dom ((𝑀 Sat 𝐸)‘𝑁) ↔ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(1st𝑢) = 𝑓))
202197, 201bitrd 282 . . . . . . . . 9 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑓 ∈ (Fmla‘𝑁) ↔ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(1st𝑢) = 𝑓))
203 r19.41v 3303 . . . . . . . . . . 11 (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)((1st𝑢) = 𝑓 ∧ ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔)) ↔ (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(1st𝑢) = 𝑓 ∧ ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔)))
20442eqcomi 2810 . . . . . . . . . . . . 13 ((𝑀 Sat 𝐸)‘𝑁) = (𝑆𝑁)
205204rexeqi 3366 . . . . . . . . . . . 12 (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)((1st𝑢) = 𝑓 ∧ ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔)) ↔ ∃𝑢 ∈ (𝑆𝑁)((1st𝑢) = 𝑓 ∧ ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔)))
206158rexbidv 3259 . . . . . . . . . . . . . . . 16 ((1st𝑢) = 𝑓 → (∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔)))
207206adantl 485 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ (1st𝑢) = 𝑓) → (∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = ((1st𝑢)⊼𝑔𝑔) ↔ ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔)))
208146eleq2d 2878 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)) ↔ 𝑔 ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁))))
209 releldmdifi 7730 . . . . . . . . . . . . . . . . . . . 20 ((Rel ((𝑀 Sat 𝐸)‘suc 𝑁) ∧ ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)) → (𝑔 ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁)) → ∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑣) = 𝑔))
21080, 150, 209syl2anc 587 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑔 ∈ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∖ dom ((𝑀 Sat 𝐸)‘𝑁)) → ∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑣) = 𝑔))
211208, 210sylbid 243 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)) → ∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑣) = 𝑔))
212154rexeqi 3366 . . . . . . . . . . . . . . . . . . 19 (∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑣) = 𝑔 ↔ ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(1st𝑣) = 𝑔)
213178biimpcd 252 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = ((1st𝑢)⊼𝑔𝑔) → ((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
214213adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑥 = ((1st𝑢)⊼𝑔𝑔)) → ((1st𝑣) = 𝑔𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
215214reximdv 3235 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ 𝑥 = ((1st𝑢)⊼𝑔𝑔)) → (∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(1st𝑣) = 𝑔 → ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
216215ex 416 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑥 = ((1st𝑢)⊼𝑔𝑔) → (∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(1st𝑣) = 𝑔 → ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))))
217216com23 86 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(1st𝑣) = 𝑔 → (𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))))
218212, 217syl5bi 245 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(1st𝑣) = 𝑔 → (𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))))
219211, 218syld 47 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁)) → (𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))))
220219rexlimdv 3245 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
221220adantr 484 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ (1st𝑢) = 𝑓) → (∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = ((1st𝑢)⊼𝑔𝑔) → ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
222207, 221sylbird 263 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) ∧ (1st𝑢) = 𝑓) → (∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔) → ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
223222expimpd 457 . . . . . . . . . . . . 13 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (((1st𝑢) = 𝑓 ∧ ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔)) → ∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
224223reximdv 3235 . . . . . . . . . . . 12 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑢 ∈ (𝑆𝑁)((1st𝑢) = 𝑓 ∧ ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔)) → ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
225205, 224syl5bi 245 . . . . . . . . . . 11 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)((1st𝑢) = 𝑓 ∧ ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔)) → ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
226203, 225syl5bir 246 . . . . . . . . . 10 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(1st𝑢) = 𝑓 ∧ ∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔)) → ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
227226expd 419 . . . . . . . . 9 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)(1st𝑢) = 𝑓 → (∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔) → ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))))
228202, 227sylbid 243 . . . . . . . 8 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (𝑓 ∈ (Fmla‘𝑁) → (∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔) → ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))))
229228rexlimdv 3245 . . . . . . 7 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (∃𝑓 ∈ (Fmla‘𝑁)∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔) → ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))))
230196, 229orim12d 962 . . . . . 6 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((∃𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ∨ ∃𝑓 ∈ (Fmla‘𝑁)∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔)) → (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))))
231141, 230impbid 215 . . . . 5 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣))) ↔ (∃𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ∨ ∃𝑓 ∈ (Fmla‘𝑁)∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔))))
232231abbidv 2865 . . . 4 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → {𝑥 ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st𝑢)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)))} = {𝑥 ∣ (∃𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ∨ ∃𝑓 ∈ (Fmla‘𝑁)∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔))})
23333, 232eqtrd 2836 . . 3 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴))} = {𝑥 ∣ (∃𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ∨ ∃𝑓 ∈ (Fmla‘𝑁)∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔))})
23412, 233ineq12d 4143 . 2 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (dom (𝑆‘suc 𝑁) ∩ dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴))}) = ((Fmla‘suc 𝑁) ∩ {𝑥 ∣ (∃𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ∨ ∃𝑓 ∈ (Fmla‘𝑁)∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔))}))
235 fmlasucdisj 32760 . . 3 (𝑁 ∈ ω → ((Fmla‘suc 𝑁) ∩ {𝑥 ∣ (∃𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ∨ ∃𝑓 ∈ (Fmla‘𝑁)∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔))}) = ∅)
236235ad2antrr 725 . 2 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → ((Fmla‘suc 𝑁) ∩ {𝑥 ∣ (∃𝑓 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))(∃𝑔 ∈ (Fmla‘suc 𝑁)𝑥 = (𝑓𝑔𝑔) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖𝑓) ∨ ∃𝑓 ∈ (Fmla‘𝑁)∃𝑔 ∈ ((Fmla‘suc 𝑁) ∖ (Fmla‘𝑁))𝑥 = (𝑓𝑔𝑔))}) = ∅)
237234, 236eqtrd 2836 1 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun (𝑆‘suc 𝑁)) → (dom (𝑆‘suc 𝑁) ∩ dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(∃𝑣 ∈ (𝑆‘suc 𝑁)(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = 𝐵)) ∨ ∃𝑢 ∈ (𝑆𝑁)∃𝑣 ∈ ((𝑆‘suc 𝑁) ∖ (𝑆𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = 𝐴))}) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  {cab 2779  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  cdif 3881  cun 3882  cin 3883  wss 3884  c0 4246  {csn 4528  cop 4534  {copab 5095  dom cdm 5523  cres 5525  Rel wrel 5528  suc csuc 6165  Fun wfun 6322  cfv 6328  (class class class)co 7139  ωcom 7564  1st c1st 7673  2nd c2nd 7674  m cmap 8393  𝑔cgna 32695  𝑔cgol 32696   Sat csat 32697  Fmlacfmla 32698
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  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 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-map 8395  df-goel 32701  df-gona 32702  df-goal 32703  df-sat 32704  df-fmla 32706
This theorem is referenced by:  satffunlem2  32769
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