Metamath Proof Explorer


Theorem satffunlem1lem1

Description: Lemma for satffunlem1 . (Contributed by AV, 17-Oct-2023)

Ref Expression
Assertion satffunlem1lem1 ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → Fun { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) } )

Proof

Step Hyp Ref Expression
1 fveq2 ( 𝑢 = 𝑠 → ( 1st𝑢 ) = ( 1st𝑠 ) )
2 fveq2 ( 𝑣 = 𝑟 → ( 1st𝑣 ) = ( 1st𝑟 ) )
3 1 2 oveqan12d ( ( 𝑢 = 𝑠𝑣 = 𝑟 ) → ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) )
4 3 eqeq2d ( ( 𝑢 = 𝑠𝑣 = 𝑟 ) → ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ↔ 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ) )
5 fveq2 ( 𝑢 = 𝑠 → ( 2nd𝑢 ) = ( 2nd𝑠 ) )
6 fveq2 ( 𝑣 = 𝑟 → ( 2nd𝑣 ) = ( 2nd𝑟 ) )
7 5 6 ineqan12d ( ( 𝑢 = 𝑠𝑣 = 𝑟 ) → ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) = ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) )
8 7 difeq2d ( ( 𝑢 = 𝑠𝑣 = 𝑟 ) → ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) )
9 8 eqeq2d ( ( 𝑢 = 𝑠𝑣 = 𝑟 ) → ( 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ↔ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) )
10 4 9 anbi12d ( ( 𝑢 = 𝑠𝑣 = 𝑟 ) → ( ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ↔ ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) ) )
11 10 cbvrexdva ( 𝑢 = 𝑠 → ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ↔ ∃ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) ) )
12 simpr ( ( 𝑢 = 𝑠𝑖 = 𝑗 ) → 𝑖 = 𝑗 )
13 1 adantr ( ( 𝑢 = 𝑠𝑖 = 𝑗 ) → ( 1st𝑢 ) = ( 1st𝑠 ) )
14 12 13 goaleq12d ( ( 𝑢 = 𝑠𝑖 = 𝑗 ) → ∀𝑔 𝑖 ( 1st𝑢 ) = ∀𝑔 𝑗 ( 1st𝑠 ) )
15 14 eqeq2d ( ( 𝑢 = 𝑠𝑖 = 𝑗 ) → ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ↔ 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ) )
16 opeq1 ( 𝑖 = 𝑗 → ⟨ 𝑖 , 𝑘 ⟩ = ⟨ 𝑗 , 𝑘 ⟩ )
17 16 sneqd ( 𝑖 = 𝑗 → { ⟨ 𝑖 , 𝑘 ⟩ } = { ⟨ 𝑗 , 𝑘 ⟩ } )
18 sneq ( 𝑖 = 𝑗 → { 𝑖 } = { 𝑗 } )
19 18 difeq2d ( 𝑖 = 𝑗 → ( ω ∖ { 𝑖 } ) = ( ω ∖ { 𝑗 } ) )
20 19 reseq2d ( 𝑖 = 𝑗 → ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) = ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) )
21 17 20 uneq12d ( 𝑖 = 𝑗 → ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) = ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) )
22 21 adantl ( ( 𝑢 = 𝑠𝑖 = 𝑗 ) → ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) = ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) )
23 5 adantr ( ( 𝑢 = 𝑠𝑖 = 𝑗 ) → ( 2nd𝑢 ) = ( 2nd𝑠 ) )
24 22 23 eleq12d ( ( 𝑢 = 𝑠𝑖 = 𝑗 ) → ( ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) ↔ ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) ) )
25 24 ralbidv ( ( 𝑢 = 𝑠𝑖 = 𝑗 ) → ( ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) ↔ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) ) )
26 25 rabbidv ( ( 𝑢 = 𝑠𝑖 = 𝑗 ) → { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } )
27 26 eqeq2d ( ( 𝑢 = 𝑠𝑖 = 𝑗 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ↔ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) )
28 15 27 anbi12d ( ( 𝑢 = 𝑠𝑖 = 𝑗 ) → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ↔ ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) ) )
29 28 cbvrexdva ( 𝑢 = 𝑠 → ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ↔ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) ) )
30 11 29 orbi12d ( 𝑢 = 𝑠 → ( ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) ↔ ( ∃ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) ) ) )
31 30 cbvrexvw ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) ↔ ∃ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) ) )
32 simp-4l ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) )
33 simpr ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) )
34 33 anim1i ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) )
35 simpr ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) )
36 35 anim1i ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) )
37 36 ad2antrr ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) )
38 satffunlem ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ∧ ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∧ ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) ) → 𝑧 = 𝑦 )
39 38 eqcomd ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ∧ ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∧ ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) ) → 𝑦 = 𝑧 )
40 39 3exp ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) → ( ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) )
41 32 34 37 40 syl3anc ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) → ( ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) )
42 41 rexlimdva ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) → ( ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) )
43 eqeq1 ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) → ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ↔ ∀𝑔 𝑖 ( 1st𝑢 ) = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ) )
44 df-goal 𝑔 𝑖 ( 1st𝑢 ) = ⟨ 2o , ⟨ 𝑖 , ( 1st𝑢 ) ⟩ ⟩
45 fvex ( 1st𝑠 ) ∈ V
46 fvex ( 1st𝑟 ) ∈ V
47 gonafv ( ( ( 1st𝑠 ) ∈ V ∧ ( 1st𝑟 ) ∈ V ) → ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) = ⟨ 1o , ⟨ ( 1st𝑠 ) , ( 1st𝑟 ) ⟩ ⟩ )
48 45 46 47 mp2an ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) = ⟨ 1o , ⟨ ( 1st𝑠 ) , ( 1st𝑟 ) ⟩ ⟩
49 44 48 eqeq12i ( ∀𝑔 𝑖 ( 1st𝑢 ) = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ↔ ⟨ 2o , ⟨ 𝑖 , ( 1st𝑢 ) ⟩ ⟩ = ⟨ 1o , ⟨ ( 1st𝑠 ) , ( 1st𝑟 ) ⟩ ⟩ )
50 2oex 2o ∈ V
51 opex 𝑖 , ( 1st𝑢 ) ⟩ ∈ V
52 50 51 opth ( ⟨ 2o , ⟨ 𝑖 , ( 1st𝑢 ) ⟩ ⟩ = ⟨ 1o , ⟨ ( 1st𝑠 ) , ( 1st𝑟 ) ⟩ ⟩ ↔ ( 2o = 1o ∧ ⟨ 𝑖 , ( 1st𝑢 ) ⟩ = ⟨ ( 1st𝑠 ) , ( 1st𝑟 ) ⟩ ) )
53 1one2o 1o ≠ 2o
54 df-ne ( 1o ≠ 2o ↔ ¬ 1o = 2o )
55 pm2.21 ( ¬ 1o = 2o → ( 1o = 2o → ( 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) → 𝑦 = 𝑧 ) ) )
56 54 55 sylbi ( 1o ≠ 2o → ( 1o = 2o → ( 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) → 𝑦 = 𝑧 ) ) )
57 53 56 ax-mp ( 1o = 2o → ( 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) → 𝑦 = 𝑧 ) )
58 57 eqcoms ( 2o = 1o → ( 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) → 𝑦 = 𝑧 ) )
59 58 adantr ( ( 2o = 1o ∧ ⟨ 𝑖 , ( 1st𝑢 ) ⟩ = ⟨ ( 1st𝑠 ) , ( 1st𝑟 ) ⟩ ) → ( 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) → 𝑦 = 𝑧 ) )
60 52 59 sylbi ( ⟨ 2o , ⟨ 𝑖 , ( 1st𝑢 ) ⟩ ⟩ = ⟨ 1o , ⟨ ( 1st𝑠 ) , ( 1st𝑟 ) ⟩ ⟩ → ( 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) → 𝑦 = 𝑧 ) )
61 49 60 sylbi ( ∀𝑔 𝑖 ( 1st𝑢 ) = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) → ( 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) → 𝑦 = 𝑧 ) )
62 43 61 syl6bi ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) → ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) → ( 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) → 𝑦 = 𝑧 ) ) )
63 62 impd ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) → ( ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) → 𝑦 = 𝑧 ) )
64 63 adantr ( ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) → ( ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) → 𝑦 = 𝑧 ) )
65 64 a1i ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) → ( ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) )
66 65 rexlimdva ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) → ( ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) )
67 42 66 jaod ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) → ( ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) )
68 67 rexlimdva ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) → ( ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) )
69 68 com23 ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) )
70 69 rexlimdva ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) )
71 eqeq1 ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) → ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ↔ ∀𝑔 𝑗 ( 1st𝑠 ) = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ) )
72 df-goal 𝑔 𝑗 ( 1st𝑠 ) = ⟨ 2o , ⟨ 𝑗 , ( 1st𝑠 ) ⟩ ⟩
73 fvex ( 1st𝑢 ) ∈ V
74 fvex ( 1st𝑣 ) ∈ V
75 gonafv ( ( ( 1st𝑢 ) ∈ V ∧ ( 1st𝑣 ) ∈ V ) → ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) = ⟨ 1o , ⟨ ( 1st𝑢 ) , ( 1st𝑣 ) ⟩ ⟩ )
76 73 74 75 mp2an ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) = ⟨ 1o , ⟨ ( 1st𝑢 ) , ( 1st𝑣 ) ⟩ ⟩
77 72 76 eqeq12i ( ∀𝑔 𝑗 ( 1st𝑠 ) = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ↔ ⟨ 2o , ⟨ 𝑗 , ( 1st𝑠 ) ⟩ ⟩ = ⟨ 1o , ⟨ ( 1st𝑢 ) , ( 1st𝑣 ) ⟩ ⟩ )
78 opex 𝑗 , ( 1st𝑠 ) ⟩ ∈ V
79 50 78 opth ( ⟨ 2o , ⟨ 𝑗 , ( 1st𝑠 ) ⟩ ⟩ = ⟨ 1o , ⟨ ( 1st𝑢 ) , ( 1st𝑣 ) ⟩ ⟩ ↔ ( 2o = 1o ∧ ⟨ 𝑗 , ( 1st𝑠 ) ⟩ = ⟨ ( 1st𝑢 ) , ( 1st𝑣 ) ⟩ ) )
80 pm2.21 ( ¬ 1o = 2o → ( 1o = 2o𝑦 = 𝑧 ) )
81 54 80 sylbi ( 1o ≠ 2o → ( 1o = 2o𝑦 = 𝑧 ) )
82 53 81 ax-mp ( 1o = 2o𝑦 = 𝑧 )
83 82 eqcoms ( 2o = 1o𝑦 = 𝑧 )
84 83 adantr ( ( 2o = 1o ∧ ⟨ 𝑗 , ( 1st𝑠 ) ⟩ = ⟨ ( 1st𝑢 ) , ( 1st𝑣 ) ⟩ ) → 𝑦 = 𝑧 )
85 79 84 sylbi ( ⟨ 2o , ⟨ 𝑗 , ( 1st𝑠 ) ⟩ ⟩ = ⟨ 1o , ⟨ ( 1st𝑢 ) , ( 1st𝑣 ) ⟩ ⟩ → 𝑦 = 𝑧 )
86 77 85 sylbi ( ∀𝑔 𝑗 ( 1st𝑠 ) = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) → 𝑦 = 𝑧 )
87 71 86 syl6bi ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) → ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) → 𝑦 = 𝑧 ) )
88 87 adantr ( ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) → 𝑦 = 𝑧 ) )
89 88 com12 ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → 𝑦 = 𝑧 ) )
90 89 adantr ( ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → 𝑦 = 𝑧 ) )
91 90 a1i ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → 𝑦 = 𝑧 ) ) )
92 91 rexlimdva ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → 𝑦 = 𝑧 ) ) )
93 eqeq1 ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) → ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ↔ ∀𝑔 𝑖 ( 1st𝑢 ) = ∀𝑔 𝑗 ( 1st𝑠 ) ) )
94 44 72 eqeq12i ( ∀𝑔 𝑖 ( 1st𝑢 ) = ∀𝑔 𝑗 ( 1st𝑠 ) ↔ ⟨ 2o , ⟨ 𝑖 , ( 1st𝑢 ) ⟩ ⟩ = ⟨ 2o , ⟨ 𝑗 , ( 1st𝑠 ) ⟩ ⟩ )
95 50 51 opth ( ⟨ 2o , ⟨ 𝑖 , ( 1st𝑢 ) ⟩ ⟩ = ⟨ 2o , ⟨ 𝑗 , ( 1st𝑠 ) ⟩ ⟩ ↔ ( 2o = 2o ∧ ⟨ 𝑖 , ( 1st𝑢 ) ⟩ = ⟨ 𝑗 , ( 1st𝑠 ) ⟩ ) )
96 vex 𝑖 ∈ V
97 96 73 opth ( ⟨ 𝑖 , ( 1st𝑢 ) ⟩ = ⟨ 𝑗 , ( 1st𝑠 ) ⟩ ↔ ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) )
98 97 anbi2i ( ( 2o = 2o ∧ ⟨ 𝑖 , ( 1st𝑢 ) ⟩ = ⟨ 𝑗 , ( 1st𝑠 ) ⟩ ) ↔ ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) ) )
99 94 95 98 3bitri ( ∀𝑔 𝑖 ( 1st𝑢 ) = ∀𝑔 𝑗 ( 1st𝑠 ) ↔ ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) ) )
100 93 99 bitrdi ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) → ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ↔ ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) ) ) )
101 100 adantl ( ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ) → ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ↔ ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) ) ) )
102 funfv1st2nd ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑠 ) )
103 102 ex ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑠 ) ) )
104 funfv1st2nd ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑢 ) ) = ( 2nd𝑢 ) )
105 104 ex ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑢 ) ) = ( 2nd𝑢 ) ) )
106 fveqeq2 ( ( 1st𝑢 ) = ( 1st𝑠 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑢 ) ) = ( 2nd𝑢 ) ↔ ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑢 ) ) )
107 eqtr2 ( ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑢 ) ∧ ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑠 ) ) → ( 2nd𝑢 ) = ( 2nd𝑠 ) )
108 opeq1 ( 𝑗 = 𝑖 → ⟨ 𝑗 , 𝑘 ⟩ = ⟨ 𝑖 , 𝑘 ⟩ )
109 108 sneqd ( 𝑗 = 𝑖 → { ⟨ 𝑗 , 𝑘 ⟩ } = { ⟨ 𝑖 , 𝑘 ⟩ } )
110 sneq ( 𝑗 = 𝑖 → { 𝑗 } = { 𝑖 } )
111 110 difeq2d ( 𝑗 = 𝑖 → ( ω ∖ { 𝑗 } ) = ( ω ∖ { 𝑖 } ) )
112 111 reseq2d ( 𝑗 = 𝑖 → ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) = ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) )
113 109 112 uneq12d ( 𝑗 = 𝑖 → ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) = ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) )
114 113 eqcoms ( 𝑖 = 𝑗 → ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) = ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) )
115 114 adantl ( ( ( 2nd𝑢 ) = ( 2nd𝑠 ) ∧ 𝑖 = 𝑗 ) → ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) = ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) )
116 simpl ( ( ( 2nd𝑢 ) = ( 2nd𝑠 ) ∧ 𝑖 = 𝑗 ) → ( 2nd𝑢 ) = ( 2nd𝑠 ) )
117 116 eqcomd ( ( ( 2nd𝑢 ) = ( 2nd𝑠 ) ∧ 𝑖 = 𝑗 ) → ( 2nd𝑠 ) = ( 2nd𝑢 ) )
118 115 117 eleq12d ( ( ( 2nd𝑢 ) = ( 2nd𝑠 ) ∧ 𝑖 = 𝑗 ) → ( ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) ↔ ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) ) )
119 118 ralbidv ( ( ( 2nd𝑢 ) = ( 2nd𝑠 ) ∧ 𝑖 = 𝑗 ) → ( ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) ↔ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) ) )
120 119 rabbidv ( ( ( 2nd𝑢 ) = ( 2nd𝑠 ) ∧ 𝑖 = 𝑗 ) → { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } )
121 eqeq12 ( ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) → ( 𝑦 = 𝑧 ↔ { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) )
122 120 121 syl5ibrcom ( ( ( 2nd𝑢 ) = ( 2nd𝑠 ) ∧ 𝑖 = 𝑗 ) → ( ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) → 𝑦 = 𝑧 ) )
123 122 exp4b ( ( 2nd𝑢 ) = ( 2nd𝑠 ) → ( 𝑖 = 𝑗 → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) )
124 107 123 syl ( ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑢 ) ∧ ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑠 ) ) → ( 𝑖 = 𝑗 → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) )
125 124 ex ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑢 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑠 ) → ( 𝑖 = 𝑗 → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) )
126 106 125 syl6bi ( ( 1st𝑢 ) = ( 1st𝑠 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑢 ) ) = ( 2nd𝑢 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑠 ) → ( 𝑖 = 𝑗 → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) )
127 126 com24 ( ( 1st𝑢 ) = ( 1st𝑠 ) → ( 𝑖 = 𝑗 → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑠 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑢 ) ) = ( 2nd𝑢 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) )
128 127 impcom ( ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑠 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑢 ) ) = ( 2nd𝑢 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) )
129 128 com13 ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑢 ) ) = ( 2nd𝑢 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑠 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) )
130 105 129 syl6 ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑠 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) )
131 130 com23 ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st𝑠 ) ) = ( 2nd𝑠 ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) )
132 103 131 syld ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) )
133 132 imp ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) )
134 133 adantr ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) )
135 134 imp ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) )
136 135 adantld ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) )
137 136 ad2antrr ( ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ) → ( ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st𝑢 ) = ( 1st𝑠 ) ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) )
138 101 137 sylbid ( ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ) → ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) )
139 138 impd ( ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) )
140 139 ex ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) → ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → 𝑦 = 𝑧 ) ) ) )
141 140 com34 ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) → ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) → ( 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → 𝑦 = 𝑧 ) ) ) )
142 141 impd ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → 𝑦 = 𝑧 ) ) )
143 142 rexlimdva ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → 𝑦 = 𝑧 ) ) )
144 92 143 jaod ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → 𝑦 = 𝑧 ) ) )
145 144 rexlimdva ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → 𝑦 = 𝑧 ) ) )
146 145 com23 ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) )
147 146 rexlimdva ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) )
148 70 147 jaod ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( ∃ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) )
149 148 rexlimdva ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ∃ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑠 ) ⊼𝑔 ( 1st𝑟 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑠 ) ∩ ( 2nd𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑗 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd𝑠 ) } ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) )
150 31 149 syl5bi ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) )
151 150 impd ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) ∧ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) ) → 𝑦 = 𝑧 ) )
152 151 alrimivv ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ∀ 𝑦𝑧 ( ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) ∧ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) ) → 𝑦 = 𝑧 ) )
153 eqeq1 ( 𝑦 = 𝑧 → ( 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ↔ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) )
154 153 anbi2d ( 𝑦 = 𝑧 → ( ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ↔ ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ) )
155 154 rexbidv ( 𝑦 = 𝑧 → ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ↔ ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ) )
156 eqeq1 ( 𝑦 = 𝑧 → ( 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ↔ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) )
157 156 anbi2d ( 𝑦 = 𝑧 → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ↔ ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) )
158 157 rexbidv ( 𝑦 = 𝑧 → ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ↔ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) )
159 155 158 orbi12d ( 𝑦 = 𝑧 → ( ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) ↔ ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) ) )
160 159 rexbidv ( 𝑦 = 𝑧 → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) ↔ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) ) )
161 160 mo4 ( ∃* 𝑦𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) ↔ ∀ 𝑦𝑧 ( ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) ∧ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑧 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) ) → 𝑦 = 𝑧 ) )
162 152 161 sylibr ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ∃* 𝑦𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) )
163 162 alrimiv ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ∀ 𝑥 ∃* 𝑦𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) )
164 funopab ( Fun { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) } ↔ ∀ 𝑥 ∃* 𝑦𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) )
165 163 164 sylibr ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → Fun { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st𝑢 ) ⊼𝑔 ( 1st𝑣 ) ) ∧ 𝑦 = ( ( 𝑀m ω ) ∖ ( ( 2nd𝑢 ) ∩ ( 2nd𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀m ω ) ∣ ∀ 𝑘𝑀 ( { ⟨ 𝑖 , 𝑘 ⟩ } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd𝑢 ) } ) ) } )