| Step |
Hyp |
Ref |
Expression |
| 1 |
|
simpll |
⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) ) → Fun 𝐹 ) |
| 2 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) ⊆ dom 𝐹 |
| 3 |
2
|
sseli |
⊢ ( 𝑦 ∈ ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) → 𝑦 ∈ dom 𝐹 ) |
| 4 |
3
|
adantl |
⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑦 ∈ dom 𝐹 ) |
| 5 |
|
fvex |
⊢ ( 𝐹 ‘ 𝑦 ) ∈ V |
| 6 |
|
fvimacnvi |
⊢ ( ( Fun 𝐹 ∧ 𝑦 ∈ ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ) |
| 7 |
6
|
adantlr |
⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ) |
| 8 |
|
eliin |
⊢ ( ( 𝐹 ‘ 𝑦 ) ∈ V → ( ( 𝐹 ‘ 𝑦 ) ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ) ) |
| 9 |
8
|
biimpa |
⊢ ( ( ( 𝐹 ‘ 𝑦 ) ∈ V ∧ ( 𝐹 ‘ 𝑦 ) ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ) → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ) |
| 10 |
5 7 9
|
sylancr |
⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) ) → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ) |
| 11 |
|
fvimacnv |
⊢ ( ( Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ↔ 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) ) ) |
| 12 |
11
|
ralbidv |
⊢ ( ( Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹 ) → ( ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) ) ) |
| 13 |
12
|
biimpa |
⊢ ( ( ( Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹 ) ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ) → ∀ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) ) |
| 14 |
1 4 10 13
|
syl21anc |
⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) ) → ∀ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) ) |
| 15 |
|
eliin |
⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) ↔ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) ) ) |
| 16 |
15
|
elv |
⊢ ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) ↔ ∀ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) ) |
| 17 |
14 16
|
sylibr |
⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) ∧ 𝑦 ∈ ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) ) → 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) ) |
| 18 |
|
simpll |
⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) ) → Fun 𝐹 ) |
| 19 |
15
|
biimpd |
⊢ ( 𝑦 ∈ V → ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) → ∀ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) ) ) |
| 20 |
19
|
elv |
⊢ ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) → ∀ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) ) |
| 21 |
20
|
adantl |
⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) ) → ∀ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) ) |
| 22 |
|
fvimacnvi |
⊢ ( ( Fun 𝐹 ∧ 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ) |
| 23 |
22
|
ex |
⊢ ( Fun 𝐹 → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ) ) |
| 24 |
23
|
ralimdv |
⊢ ( Fun 𝐹 → ( ∀ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ) ) |
| 25 |
18 21 24
|
sylc |
⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) ) → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ) |
| 26 |
5 8
|
ax-mp |
⊢ ( ( 𝐹 ‘ 𝑦 ) ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ) |
| 27 |
25 26
|
sylibr |
⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ) |
| 28 |
|
r19.2zb |
⊢ ( 𝐴 ≠ ∅ ↔ ( ∀ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) ) ) |
| 29 |
28
|
biimpi |
⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) ) ) |
| 30 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ 𝐵 ) ⊆ dom 𝐹 |
| 31 |
30
|
sseli |
⊢ ( 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) → 𝑦 ∈ dom 𝐹 ) |
| 32 |
31
|
rexlimivw |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) → 𝑦 ∈ dom 𝐹 ) |
| 33 |
29 32
|
syl6 |
⊢ ( 𝐴 ≠ ∅ → ( ∀ 𝑥 ∈ 𝐴 𝑦 ∈ ( ◡ 𝐹 “ 𝐵 ) → 𝑦 ∈ dom 𝐹 ) ) |
| 34 |
16 33
|
syl5bi |
⊢ ( 𝐴 ≠ ∅ → ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) → 𝑦 ∈ dom 𝐹 ) ) |
| 35 |
34
|
adantl |
⊢ ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) → ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) → 𝑦 ∈ dom 𝐹 ) ) |
| 36 |
35
|
imp |
⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) ) → 𝑦 ∈ dom 𝐹 ) |
| 37 |
|
fvimacnv |
⊢ ( ( Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝑦 ) ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) ) ) |
| 38 |
18 36 37
|
syl2anc |
⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑦 ) ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) ) ) |
| 39 |
27 38
|
mpbid |
⊢ ( ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) ) → 𝑦 ∈ ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) ) |
| 40 |
17 39
|
impbida |
⊢ ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) → ( 𝑦 ∈ ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) ↔ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) ) ) |
| 41 |
40
|
eqrdv |
⊢ ( ( Fun 𝐹 ∧ 𝐴 ≠ ∅ ) → ( ◡ 𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵 ) = ∩ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ 𝐵 ) ) |