Step |
Hyp |
Ref |
Expression |
1 |
|
marypha2lem.t |
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ( { 𝑥 } × ( 𝐹 ‘ 𝑥 ) ) |
2 |
|
dffn5 |
⊢ ( 𝐺 Fn 𝐴 ↔ 𝐺 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) ) |
3 |
2
|
biimpi |
⊢ ( 𝐺 Fn 𝐴 → 𝐺 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) ) |
4 |
3
|
adantl |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → 𝐺 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) ) |
5 |
|
df-mpt |
⊢ ( 𝑥 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = ( 𝐺 ‘ 𝑥 ) ) } |
6 |
4 5
|
eqtrdi |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → 𝐺 = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = ( 𝐺 ‘ 𝑥 ) ) } ) |
7 |
1
|
marypha2lem2 |
⊢ 𝑇 = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } |
8 |
7
|
a1i |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → 𝑇 = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ) |
9 |
6 8
|
sseq12d |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( 𝐺 ⊆ 𝑇 ↔ { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = ( 𝐺 ‘ 𝑥 ) ) } ⊆ { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ) ) |
10 |
|
ssopab2bw |
⊢ ( { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = ( 𝐺 ‘ 𝑥 ) ) } ⊆ { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = ( 𝐺 ‘ 𝑥 ) ) → ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ) |
11 |
9 10
|
bitrdi |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( 𝐺 ⊆ 𝑇 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = ( 𝐺 ‘ 𝑥 ) ) → ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ) ) |
12 |
|
19.21v |
⊢ ( ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 = ( 𝐺 ‘ 𝑥 ) → 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 = ( 𝐺 ‘ 𝑥 ) → 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ) |
13 |
|
imdistan |
⊢ ( ( 𝑥 ∈ 𝐴 → ( 𝑦 = ( 𝐺 ‘ 𝑥 ) → 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = ( 𝐺 ‘ 𝑥 ) ) → ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ) |
14 |
13
|
albii |
⊢ ( ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 = ( 𝐺 ‘ 𝑥 ) → 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ↔ ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = ( 𝐺 ‘ 𝑥 ) ) → ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ) |
15 |
|
fvex |
⊢ ( 𝐺 ‘ 𝑥 ) ∈ V |
16 |
|
eleq1 |
⊢ ( 𝑦 = ( 𝐺 ‘ 𝑥 ) → ( 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐺 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
17 |
15 16
|
ceqsalv |
⊢ ( ∀ 𝑦 ( 𝑦 = ( 𝐺 ‘ 𝑥 ) → 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ↔ ( 𝐺 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) |
18 |
17
|
imbi2i |
⊢ ( ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 = ( 𝐺 ‘ 𝑥 ) → 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝐺 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
19 |
12 14 18
|
3bitr3i |
⊢ ( ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = ( 𝐺 ‘ 𝑥 ) ) → ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝐺 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
20 |
19
|
albii |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = ( 𝐺 ‘ 𝑥 ) ) → ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝐺 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
21 |
|
df-ral |
⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝐺 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝐺 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
22 |
20 21
|
bitr4i |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = ( 𝐺 ‘ 𝑥 ) ) → ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐺 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) |
23 |
11 22
|
bitrdi |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ) → ( 𝐺 ⊆ 𝑇 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐺 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) ) |