Step |
Hyp |
Ref |
Expression |
1 |
|
satefv |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) → ( 𝑀 Sat∈ 𝑈 ) = ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) |
2 |
1
|
eleq2d |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) → ( 𝑆 ∈ ( 𝑀 Sat∈ 𝑈 ) ↔ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) ) |
3 |
|
simpl |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) → 𝑀 ∈ 𝑉 ) |
4 |
|
incom |
⊢ ( E ∩ ( 𝑀 × 𝑀 ) ) = ( ( 𝑀 × 𝑀 ) ∩ E ) |
5 |
|
sqxpexg |
⊢ ( 𝑀 ∈ 𝑉 → ( 𝑀 × 𝑀 ) ∈ V ) |
6 |
5
|
adantr |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) → ( 𝑀 × 𝑀 ) ∈ V ) |
7 |
|
inex1g |
⊢ ( ( 𝑀 × 𝑀 ) ∈ V → ( ( 𝑀 × 𝑀 ) ∩ E ) ∈ V ) |
8 |
6 7
|
syl |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) → ( ( 𝑀 × 𝑀 ) ∩ E ) ∈ V ) |
9 |
4 8
|
eqeltrid |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) → ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) |
10 |
3 9
|
jca |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) → ( 𝑀 ∈ 𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) ) |
11 |
10
|
adantr |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) → ( 𝑀 ∈ 𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) ) |
12 |
|
simpr |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) → 𝑈 ∈ ( Fmla ‘ ω ) ) |
13 |
12
|
adantr |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) → 𝑈 ∈ ( Fmla ‘ ω ) ) |
14 |
|
simpr |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) → 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) |
15 |
11 13 14
|
3jca |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) → ( ( 𝑀 ∈ 𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) ∧ 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) ) |
16 |
15
|
ex |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) → ( 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) → ( ( 𝑀 ∈ 𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) ∧ 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) ) ) |
17 |
2 16
|
sylbid |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ) → ( 𝑆 ∈ ( 𝑀 Sat∈ 𝑈 ) → ( ( 𝑀 ∈ 𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) ∧ 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) ) ) |
18 |
17
|
3impia |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( 𝑀 Sat∈ 𝑈 ) ) → ( ( 𝑀 ∈ 𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) ∧ 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) ) |
19 |
|
satfvel |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ ( E ∩ ( 𝑀 × 𝑀 ) ) ∈ V ) ∧ 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( ( ( 𝑀 Sat ( E ∩ ( 𝑀 × 𝑀 ) ) ) ‘ ω ) ‘ 𝑈 ) ) → 𝑆 : ω ⟶ 𝑀 ) |
20 |
18 19
|
syl |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝑈 ∈ ( Fmla ‘ ω ) ∧ 𝑆 ∈ ( 𝑀 Sat∈ 𝑈 ) ) → 𝑆 : ω ⟶ 𝑀 ) |