Step |
Hyp |
Ref |
Expression |
1 |
|
curry2ima.1 |
⊢ 𝐺 = ( 𝐹 ∘ ◡ ( 1st ↾ ( V × { 𝐶 } ) ) ) |
2 |
|
simp1 |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → 𝐹 Fn ( 𝐴 × 𝐵 ) ) |
3 |
|
dffn2 |
⊢ ( 𝐹 Fn ( 𝐴 × 𝐵 ) ↔ 𝐹 : ( 𝐴 × 𝐵 ) ⟶ V ) |
4 |
2 3
|
sylib |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → 𝐹 : ( 𝐴 × 𝐵 ) ⟶ V ) |
5 |
|
simp2 |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → 𝐶 ∈ 𝐵 ) |
6 |
1
|
curry2f |
⊢ ( ( 𝐹 : ( 𝐴 × 𝐵 ) ⟶ V ∧ 𝐶 ∈ 𝐵 ) → 𝐺 : 𝐴 ⟶ V ) |
7 |
4 5 6
|
syl2anc |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → 𝐺 : 𝐴 ⟶ V ) |
8 |
7
|
ffund |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → Fun 𝐺 ) |
9 |
|
simp3 |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → 𝐷 ⊆ 𝐴 ) |
10 |
7
|
fdmd |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → dom 𝐺 = 𝐴 ) |
11 |
9 10
|
sseqtrrd |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → 𝐷 ⊆ dom 𝐺 ) |
12 |
|
dfimafn |
⊢ ( ( Fun 𝐺 ∧ 𝐷 ⊆ dom 𝐺 ) → ( 𝐺 “ 𝐷 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐷 ( 𝐺 ‘ 𝑥 ) = 𝑦 } ) |
13 |
8 11 12
|
syl2anc |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → ( 𝐺 “ 𝐷 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐷 ( 𝐺 ‘ 𝑥 ) = 𝑦 } ) |
14 |
1
|
curry2val |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ) → ( 𝐺 ‘ 𝑥 ) = ( 𝑥 𝐹 𝐶 ) ) |
15 |
14
|
3adant3 |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) = ( 𝑥 𝐹 𝐶 ) ) |
16 |
15
|
eqeq1d |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → ( ( 𝐺 ‘ 𝑥 ) = 𝑦 ↔ ( 𝑥 𝐹 𝐶 ) = 𝑦 ) ) |
17 |
|
eqcom |
⊢ ( ( 𝑥 𝐹 𝐶 ) = 𝑦 ↔ 𝑦 = ( 𝑥 𝐹 𝐶 ) ) |
18 |
16 17
|
bitrdi |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → ( ( 𝐺 ‘ 𝑥 ) = 𝑦 ↔ 𝑦 = ( 𝑥 𝐹 𝐶 ) ) ) |
19 |
18
|
rexbidv |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → ( ∃ 𝑥 ∈ 𝐷 ( 𝐺 ‘ 𝑥 ) = 𝑦 ↔ ∃ 𝑥 ∈ 𝐷 𝑦 = ( 𝑥 𝐹 𝐶 ) ) ) |
20 |
19
|
abbidv |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐷 ( 𝐺 ‘ 𝑥 ) = 𝑦 } = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐷 𝑦 = ( 𝑥 𝐹 𝐶 ) } ) |
21 |
13 20
|
eqtrd |
⊢ ( ( 𝐹 Fn ( 𝐴 × 𝐵 ) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴 ) → ( 𝐺 “ 𝐷 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐷 𝑦 = ( 𝑥 𝐹 𝐶 ) } ) |