Theorem satffunlem2lem2

Description: Lemma 2 for satffunlem2 . (Contributed by AV, 27-Oct-2023)

	Ref	Expression
Hypotheses	satffunlem2lem2.s	⊢ 𝑆 = ( 𝑀 Sat 𝐸 )
	satffunlem2lem2.a	⊢ 𝐴 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) )
	satffunlem2lem2.b	⊢ 𝐵 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) }
Assertion	satffunlem2lem2	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( dom ( 𝑆 ‘ suc 𝑁 ) ∩ dom { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		satffunlem2lem2.s	⊢ 𝑆 = ( 𝑀 Sat 𝐸 )
2		satffunlem2lem2.a	⊢ 𝐴 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) )
3		satffunlem2lem2.b	⊢ 𝐵 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { ⟨ 𝑖 , 𝑧 ⟩ } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) }
4	1	fveq1i	⊢ ( 𝑆 ‘ suc 𝑁 ) = ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 )
5	4	dmeqi	⊢ dom ( 𝑆 ‘ suc 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 )
6		simprl	⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → 𝑀 ∈ 𝑉 )
7		simprr	⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → 𝐸 ∈ 𝑊 )
8		peano2	⊢ ( 𝑁 ∈ ω → suc 𝑁 ∈ ω )
9	8	adantr	⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → suc 𝑁 ∈ ω )
10	6 7 9	3jca	⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω ) )
11		satfdmfmla	⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) = ( Fmla ‘ suc 𝑁 ) )
12	10 11	syl	⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) = ( Fmla ‘ suc 𝑁 ) )
13	12	adantr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) = ( Fmla ‘ suc 𝑁 ) )
14	5 13	eqtrid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom ( 𝑆 ‘ suc 𝑁 ) = ( Fmla ‘ suc 𝑁 ) )
15		ovex	⊢ ( 𝑀 ↑m ω ) ∈ V
16	15	difexi	⊢ ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ∈ V
17	2 16	eqeltri	⊢ 𝐴 ∈ V
18	17	a1i	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → 𝐴 ∈ V )
19	18	ralrimiva	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ∀ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝐴 ∈ V )
20	3 15	rabex2	⊢ 𝐵 ∈ V
21	20	a1i	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑖 ∈ ω ) → 𝐵 ∈ V )
22	21	ralrimiva	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ∀ 𝑖 ∈ ω 𝐵 ∈ V )
23	19 22	jca	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( ∀ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝐴 ∈ V ∧ ∀ 𝑖 ∈ ω 𝐵 ∈ V ) )
24	23	ralrimiva	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ∀ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∀ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝐴 ∈ V ∧ ∀ 𝑖 ∈ ω 𝐵 ∈ V ) )
25		simplr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) )
26	8	ancri	⊢ ( 𝑁 ∈ ω → ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) )
27	26	ad2antrr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) )
28	25 27	jca	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) )
29		sssucid	⊢ 𝑁 ⊆ suc 𝑁
30	1	satfsschain	⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) → ( 𝑁 ⊆ suc 𝑁 → ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) )
31	28 29 30	mpisyl	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) )
32		dmopab3rexdif	⊢ ( ( ∀ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∀ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝐴 ∈ V ∧ ∀ 𝑖 ∈ ω 𝐵 ∈ V ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → dom { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } = { 𝑥 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) } )
33	24 31 32	syl2anc	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } = { 𝑥 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) } )
34		simpr	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ 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	⊢ ( 𝑤 = 𝑢 → ( ( 1st ‘ 𝑤 ) = ( 1st ‘ 𝑢 ) ↔ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑢 ) ) )
36	35	adantl	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ∧ 𝑤 = 𝑢 ) → ( ( 1st ‘ 𝑤 ) = ( 1st ‘ 𝑢 ) ↔ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑢 ) ) )
37		eqidd	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑢 ) )
38	34 36 37	rspcedvd	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ∃ 𝑤 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑤 ) = ( 1st ‘ 𝑢 ) )
39	4	funeqi	⊢ ( Fun ( 𝑆 ‘ suc 𝑁 ) ↔ Fun ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) )
40	39	bilani	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → Fun ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) )
41	1	fveq1i	⊢ ( 𝑆 ‘ 𝑁 ) = ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 )
42	31 41 4	3sstr3g	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) )
43	40 42	jca	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∧ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) )
44	43	adantr	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ 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 ) ) )
45		funeldmdif	⊢ ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∧ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) → ( ( 1st ‘ 𝑢 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ↔ ∃ 𝑤 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑤 ) = ( 1st ‘ 𝑢 ) ) )
46	44 45	syl	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( ( 1st ‘ 𝑢 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ↔ ∃ 𝑤 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑤 ) = ( 1st ‘ 𝑢 ) ) )
47	38 46	mpbird	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( 1st ‘ 𝑢 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) )
48	47	ex	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( 1st ‘ 𝑢 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) )
49	4 41	difeq12i	⊢ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) = ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) )
50	49	eleq2i	⊢ ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ↔ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) )
51	50	a1i	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ↔ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) )
52	13	eqcomd	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ suc 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) )
53		simpl	⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → 𝑁 ∈ ω )
54	6 7 53	3jca	⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) )
55		satfdmfmla	⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) = ( Fmla ‘ 𝑁 ) )
56	54 55	syl	⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) = ( Fmla ‘ 𝑁 ) )
57	56	eqcomd	⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( Fmla ‘ 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) )
58	57	adantr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) )
59	52 58	difeq12d	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) = ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) )
60	59	eleq2d	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 1st ‘ 𝑢 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ↔ ( 1st ‘ 𝑢 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) )
61	48 51 60	3imtr4d	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → ( 1st ‘ 𝑢 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) )
62	61	imp	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( 1st ‘ 𝑢 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) )
63	62	adantr	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) → ( 1st ‘ 𝑢 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) )
64		oveq1	⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( 𝑓 ⊼𝑔 𝑔 ) = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) )
65	64	eqeq2d	⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ↔ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) )
66	65	rexbidv	⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ↔ ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) )
67		eqidd	⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → 𝑖 = 𝑖 )
68		id	⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → 𝑓 = ( 1st ‘ 𝑢 ) )
69	67 68	goaleq12d	⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ∀𝑔 𝑖 𝑓 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) )
70	69	eqeq2d	⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( 𝑥 = ∀𝑔 𝑖 𝑓 ↔ 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) )
71	70	rexbidv	⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ↔ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) )
72	66 71	orbi12d	⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ↔ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) )
73	72	adantl	⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ∧ 𝑓 = ( 1st ‘ 𝑢 ) ) → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ↔ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) )
74	6	adantr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → 𝑀 ∈ 𝑉 )
75	7	adantr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → 𝐸 ∈ 𝑊 )
76	8	ad2antrr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → suc 𝑁 ∈ ω )
77	74 75 76	3jca	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω ) )
78		satfrel	⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω ) → Rel ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) )
79	77 78	syl	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → Rel ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) )
80	4	releqi	⊢ ( Rel ( 𝑆 ‘ suc 𝑁 ) ↔ Rel ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) )
81	79 80	sylibr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → Rel ( 𝑆 ‘ suc 𝑁 ) )
82		1stdm	⊢ ( ( Rel ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( 1st ‘ 𝑣 ) ∈ dom ( 𝑆 ‘ suc 𝑁 ) )
83	81 82	sylan	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( 1st ‘ 𝑣 ) ∈ dom ( 𝑆 ‘ suc 𝑁 ) )
84	14	eqcomd	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ suc 𝑁 ) = dom ( 𝑆 ‘ suc 𝑁 ) )
85	84	adantr	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ suc 𝑁 ) = dom ( 𝑆 ‘ suc 𝑁 ) )
86	83 85	eleqtrrd	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( 1st ‘ 𝑣 ) ∈ ( Fmla ‘ suc 𝑁 ) )
87	86	ad4ant13	⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ( 1st ‘ 𝑣 ) ∈ ( Fmla ‘ suc 𝑁 ) )
88		oveq2	⊢ ( 𝑔 = ( 1st ‘ 𝑣 ) → ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) )
89	88	eqeq2d	⊢ ( 𝑔 = ( 1st ‘ 𝑣 ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
90	89	adantl	⊢ ( ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ∧ 𝑔 = ( 1st ‘ 𝑣 ) ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
91		simpr	⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) )
92	87 90 91	rspcedvd	⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) )
93	92	rexlimdva2	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) → ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) )
94	93	orim1d	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) )
95	94	imp	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) → ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) )
96	63 73 95	rspcedvd	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) → ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) )
97	96	rexlimdva2	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) )
98	54	adantr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) )
99		satfrel	⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → Rel ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) )
100	98 99	syl	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → Rel ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) )
101	41	releqi	⊢ ( Rel ( 𝑆 ‘ 𝑁 ) ↔ Rel ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) )
102	100 101	sylibr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → Rel ( 𝑆 ‘ 𝑁 ) )
103		1stdm	⊢ ( ( Rel ( 𝑆 ‘ 𝑁 ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) → ( 1st ‘ 𝑢 ) ∈ dom ( 𝑆 ‘ 𝑁 ) )
104	102 103	sylan	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) → ( 1st ‘ 𝑢 ) ∈ dom ( 𝑆 ‘ 𝑁 ) )
105	41	dmeqi	⊢ dom ( 𝑆 ‘ 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 )
106	98 55	syl	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) = ( Fmla ‘ 𝑁 ) )
107	105 106	eqtrid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom ( 𝑆 ‘ 𝑁 ) = ( Fmla ‘ 𝑁 ) )
108	107	eqcomd	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ 𝑁 ) = dom ( 𝑆 ‘ 𝑁 ) )
109	108	adantr	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) → ( Fmla ‘ 𝑁 ) = dom ( 𝑆 ‘ 𝑁 ) )
110	104 109	eleqtrrd	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) → ( 1st ‘ 𝑢 ) ∈ ( Fmla ‘ 𝑁 ) )
111	110	adantr	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ( 1st ‘ 𝑢 ) ∈ ( Fmla ‘ 𝑁 ) )
112	65	rexbidv	⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ↔ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) )
113	112	adantl	⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ∧ 𝑓 = ( 1st ‘ 𝑢 ) ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ↔ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) )
114		simpr	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) )
115		fveqeq2	⊢ ( 𝑡 = 𝑣 → ( ( 1st ‘ 𝑡 ) = ( 1st ‘ 𝑣 ) ↔ ( 1st ‘ 𝑣 ) = ( 1st ‘ 𝑣 ) ) )
116	115	adantl	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ∧ 𝑡 = 𝑣 ) → ( ( 1st ‘ 𝑡 ) = ( 1st ‘ 𝑣 ) ↔ ( 1st ‘ 𝑣 ) = ( 1st ‘ 𝑣 ) ) )
117		eqidd	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( 1st ‘ 𝑣 ) = ( 1st ‘ 𝑣 ) )
118	114 116 117	rspcedvd	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ∃ 𝑡 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑡 ) = ( 1st ‘ 𝑣 ) )
119	43	adantr	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ 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 ) ) )
120		funeldmdif	⊢ ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∧ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) → ( ( 1st ‘ 𝑣 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ↔ ∃ 𝑡 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑡 ) = ( 1st ‘ 𝑣 ) ) )
121	119 120	syl	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( ( 1st ‘ 𝑣 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ↔ ∃ 𝑡 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑡 ) = ( 1st ‘ 𝑣 ) ) )
122	118 121	mpbird	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( 1st ‘ 𝑣 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) )
123	122	ex	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( 1st ‘ 𝑣 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) )
124	49	eleq2i	⊢ ( 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ↔ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) )
125	124	a1i	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ↔ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) )
126	12	eqcomd	⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( Fmla ‘ suc 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) )
127	126 57	difeq12d	⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) = ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) )
128	127	eleq2d	⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( ( 1st ‘ 𝑣 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ↔ ( 1st ‘ 𝑣 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) )
129	128	adantr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 1st ‘ 𝑣 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ↔ ( 1st ‘ 𝑣 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) )
130	123 125 129	3imtr4d	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → ( 1st ‘ 𝑣 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) )
131	130	adantr	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) → ( 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → ( 1st ‘ 𝑣 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) )
132	131	imp	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( 1st ‘ 𝑣 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) )
133	132	adantr	⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ( 1st ‘ 𝑣 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) )
134	89	adantl	⊢ ( ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ∧ 𝑔 = ( 1st ‘ 𝑣 ) ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
135		simpr	⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) )
136	133 134 135	rspcedvd	⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) )
137	136	r19.29an	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) )
138	111 113 137	rspcedvd	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) )
139	138	rexlimdva2	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) → ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) )
140	97 139	orim12d	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ) )
141	10	adantr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω ) )
142	11	eqcomd	⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω ) → ( Fmla ‘ suc 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) )
143	141 142	syl	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ suc 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) )
144	106	eqcomd	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) )
145	143 144	difeq12d	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) = ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) )
146	145	eleq2d	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ↔ 𝑓 ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) )
147		eqid	⊢ ( 𝑀 Sat 𝐸 ) = ( 𝑀 Sat 𝐸 )
148	147	satfsschain	⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) → ( 𝑁 ⊆ suc 𝑁 → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) )
149	28 29 148	mpisyl	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) )
150		releldmdifi	⊢ ( ( 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 ) ) ( 1st ‘ 𝑢 ) = 𝑓 ) )
151	79 149 150	syl2anc	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ 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 ) ) ( 1st ‘ 𝑢 ) = 𝑓 ) )
152	146 151	sylbid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) → ∃ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 ) )
153	49	eqcomi	⊢ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) = ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) )
154	153	rexeqi	⊢ ( ∃ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 ↔ ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 )
155		r19.41v	⊢ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) ↔ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 ∧ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) )
156		oveq1	⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) = ( 𝑓 ⊼𝑔 𝑔 ) )
157	156	eqeq2d	⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) )
158	157	rexbidv	⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) )
159		eqidd	⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → 𝑖 = 𝑖 )
160		id	⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( 1st ‘ 𝑢 ) = 𝑓 )
161	159 160	goaleq12d	⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = ∀𝑔 𝑖 𝑓 )
162	161	eqeq2d	⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ 𝑥 = ∀𝑔 𝑖 𝑓 ) )
163	162	rexbidv	⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) )
164	158 163	orbi12d	⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ↔ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) )
165	164	adantl	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ↔ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) )
166	141 11	syl	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) = ( Fmla ‘ suc 𝑁 ) )
167	166	eqcomd	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ suc 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) )
168	167	eleq2d	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) ↔ 𝑔 ∈ dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) )
169		releldm2	⊢ ( Rel ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) → ( 𝑔 ∈ dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ↔ ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( 1st ‘ 𝑣 ) = 𝑔 ) )
170	79 169	syl	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ↔ ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( 1st ‘ 𝑣 ) = 𝑔 ) )
171	168 170	bitrd	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) ↔ ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( 1st ‘ 𝑣 ) = 𝑔 ) )
172		r19.41v	⊢ ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) ↔ ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) )
173	1	eqcomi	⊢ ( 𝑀 Sat 𝐸 ) = 𝑆
174	173	fveq1i	⊢ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) = ( 𝑆 ‘ suc 𝑁 )
175	174	rexeqi	⊢ ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) ↔ ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) )
176	88	eqcoms	⊢ ( ( 1st ‘ 𝑣 ) = 𝑔 → ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) )
177	176	eqeq2d	⊢ ( ( 1st ‘ 𝑣 ) = 𝑔 → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
178	177	biimpa	⊢ ( ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) )
179	178	a1i	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
180	179	reximdv	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
181	175 180	biimtrid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
182	172 181	biimtrrid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
183	182	expd	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( 1st ‘ 𝑣 ) = 𝑔 → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) )
184	171 183	sylbid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) )
185	184	rexlimdv	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
186	185	ad2antrr	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
187	186	orim1d	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) )
188	165 187	sylbird	⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) )
189	188	expimpd	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) )
190	189	reximdva	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) → ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) )
191	155 190	biimtrrid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 ∧ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) → ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) )
192	191	expd	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) → ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) )
193	154 192	biimtrid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) → ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) )
194	152 193	syld	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) → ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) )
195	194	rexlimdv	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) → ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) )
196	144	eleq2d	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( Fmla ‘ 𝑁 ) ↔ 𝑓 ∈ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) )
197	54 99	syl	⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → Rel ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) )
198	197	adantr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → Rel ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) )
199		releldm2	⊢ ( Rel ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( 𝑓 ∈ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ↔ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 1st ‘ 𝑢 ) = 𝑓 ) )
200	198 199	syl	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ↔ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 1st ‘ 𝑢 ) = 𝑓 ) )
201	196 200	bitrd	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( Fmla ‘ 𝑁 ) ↔ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 1st ‘ 𝑢 ) = 𝑓 ) )
202		r19.41v	⊢ ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ↔ ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) )
203	41	eqcomi	⊢ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) = ( 𝑆 ‘ 𝑁 )
204	203	rexeqi	⊢ ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ↔ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) )
205	157	rexbidv	⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) )
206	205	adantl	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) )
207	145	eleq2d	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ↔ 𝑔 ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) )
208		releldmdifi	⊢ ( ( 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 ) ) ( 1st ‘ 𝑣 ) = 𝑔 ) )
209	79 149 208	syl2anc	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ 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 ) ) ( 1st ‘ 𝑣 ) = 𝑔 ) )
210	207 209	sylbid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) → ∃ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 ) )
211	153	rexeqi	⊢ ( ∃ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 ↔ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 )
212	177	biimpcd	⊢ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ( ( 1st ‘ 𝑣 ) = 𝑔 → 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
213	212	adantl	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → ( ( 1st ‘ 𝑣 ) = 𝑔 → 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
214	213	reximdv	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → ( ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
215	214	ex	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ( ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) )
216	215	com23	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) )
217	211 216	biimtrid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) )
218	210 217	syld	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) )
219	218	rexlimdv	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
220	219	adantr	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
221	206 220	sylbird	⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
222	221	expimpd	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
223	222	reximdv	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
224	204 223	biimtrid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
225	202 224	biimtrrid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
226	225	expd	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 1st ‘ 𝑢 ) = 𝑓 → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) )
227	201 226	sylbid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( Fmla ‘ 𝑁 ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) )
228	227	rexlimdv	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) )
229	195 228	orim12d	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) )
230	140 229	impbid	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ↔ ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ) )
231	230	abbidv	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → { 𝑥 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) } = { 𝑥 ∣ ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) } )
232	33 231	eqtrd	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } = { 𝑥 ∣ ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) } )
233	14 232	ineq12d	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( dom ( 𝑆 ‘ suc 𝑁 ) ∩ dom { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } ) = ( ( Fmla ‘ suc 𝑁 ) ∩ { 𝑥 ∣ ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) } ) )
234		fmlasucdisj	⊢ ( 𝑁 ∈ ω → ( ( Fmla ‘ suc 𝑁 ) ∩ { 𝑥 ∣ ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) } ) = ∅ )
235	234	ad2antrr	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( Fmla ‘ suc 𝑁 ) ∩ { 𝑥 ∣ ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) } ) = ∅ )
236	233 235	eqtrd	⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( dom ( 𝑆 ‘ suc 𝑁 ) ∩ dom { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } ) = ∅ )