Step |
Hyp |
Ref |
Expression |
1 |
|
marypha2lem.t |
⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ( { 𝑥 } × ( 𝐹 ‘ 𝑥 ) ) |
2 |
1
|
marypha2lem2 |
⊢ 𝑇 = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } |
3 |
2
|
imaeq1i |
⊢ ( 𝑇 “ 𝑋 ) = ( { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } “ 𝑋 ) |
4 |
|
df-ima |
⊢ ( { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } “ 𝑋 ) = ran ( { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ↾ 𝑋 ) |
5 |
3 4
|
eqtri |
⊢ ( 𝑇 “ 𝑋 ) = ran ( { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ↾ 𝑋 ) |
6 |
|
resopab2 |
⊢ ( 𝑋 ⊆ 𝐴 → ( { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ↾ 𝑋 ) = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ) |
7 |
6
|
adantl |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴 ) → ( { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ↾ 𝑋 ) = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ) |
8 |
7
|
rneqd |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴 ) → ran ( { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ↾ 𝑋 ) = ran { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ) |
9 |
|
rnopab |
⊢ ran { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } = { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } |
10 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ 𝑋 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ↔ ∃ 𝑥 ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
11 |
10
|
bicomi |
⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ↔ ∃ 𝑥 ∈ 𝑋 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) |
12 |
11
|
abbii |
⊢ { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝑋 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } |
13 |
|
df-iun |
⊢ ∪ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝑋 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) } |
14 |
12 13
|
eqtr4i |
⊢ { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } = ∪ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) |
15 |
14
|
a1i |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴 ) → { 𝑦 ∣ ∃ 𝑥 ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } = ∪ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) |
16 |
9 15
|
eqtrid |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴 ) → ran { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } = ∪ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) |
17 |
8 16
|
eqtrd |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴 ) → ran ( { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } ↾ 𝑋 ) = ∪ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) |
18 |
5 17
|
eqtrid |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑇 “ 𝑋 ) = ∪ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) ) |
19 |
|
fnfun |
⊢ ( 𝐹 Fn 𝐴 → Fun 𝐹 ) |
20 |
19
|
adantr |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴 ) → Fun 𝐹 ) |
21 |
|
funiunfv |
⊢ ( Fun 𝐹 → ∪ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) = ∪ ( 𝐹 “ 𝑋 ) ) |
22 |
20 21
|
syl |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴 ) → ∪ 𝑥 ∈ 𝑋 ( 𝐹 ‘ 𝑥 ) = ∪ ( 𝐹 “ 𝑋 ) ) |
23 |
18 22
|
eqtrd |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴 ) → ( 𝑇 “ 𝑋 ) = ∪ ( 𝐹 “ 𝑋 ) ) |