Step |
Hyp |
Ref |
Expression |
1 |
|
elex |
⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ V ) |
2 |
|
isinfcard |
⊢ ( ( ω ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( card ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ↔ ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ ) |
3 |
2
|
bicomi |
⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ ↔ ( ω ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( card ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) |
4 |
3
|
simplbi |
⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ → ω ⊆ ( 𝐹 ‘ 𝑥 ) ) |
5 |
|
ffn |
⊢ ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) → 𝐹 Fn 𝐴 ) |
6 |
|
fnfvelrn |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ ran 𝐹 ) |
7 |
6
|
ex |
⊢ ( 𝐹 Fn 𝐴 → ( 𝑥 ∈ 𝐴 → ( 𝐹 ‘ 𝑥 ) ∈ ran 𝐹 ) ) |
8 |
|
fnima |
⊢ ( 𝐹 Fn 𝐴 → ( 𝐹 “ 𝐴 ) = ran 𝐹 ) |
9 |
8
|
eleq2d |
⊢ ( 𝐹 Fn 𝐴 → ( ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 “ 𝐴 ) ↔ ( 𝐹 ‘ 𝑥 ) ∈ ran 𝐹 ) ) |
10 |
7 9
|
sylibrd |
⊢ ( 𝐹 Fn 𝐴 → ( 𝑥 ∈ 𝐴 → ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 “ 𝐴 ) ) ) |
11 |
|
elssuni |
⊢ ( ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 “ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ⊆ ∪ ( 𝐹 “ 𝐴 ) ) |
12 |
10 11
|
syl6 |
⊢ ( 𝐹 Fn 𝐴 → ( 𝑥 ∈ 𝐴 → ( 𝐹 ‘ 𝑥 ) ⊆ ∪ ( 𝐹 “ 𝐴 ) ) ) |
13 |
12
|
imp |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ⊆ ∪ ( 𝐹 “ 𝐴 ) ) |
14 |
5 13
|
sylan |
⊢ ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ⊆ ∪ ( 𝐹 “ 𝐴 ) ) |
15 |
4 14
|
sylan9ssr |
⊢ ( ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ 𝑥 ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ ) → ω ⊆ ∪ ( 𝐹 “ 𝐴 ) ) |
16 |
15
|
anasss |
⊢ ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ( 𝑥 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ ) ) → ω ⊆ ∪ ( 𝐹 “ 𝐴 ) ) |
17 |
16
|
a1i |
⊢ ( 𝐴 ∈ V → ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ( 𝑥 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ ) ) → ω ⊆ ∪ ( 𝐹 “ 𝐴 ) ) ) |
18 |
|
carduniima |
⊢ ( 𝐴 ∈ V → ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) → ∪ ( 𝐹 “ 𝐴 ) ∈ ( ω ∪ ran ℵ ) ) ) |
19 |
|
iscard3 |
⊢ ( ( card ‘ ∪ ( 𝐹 “ 𝐴 ) ) = ∪ ( 𝐹 “ 𝐴 ) ↔ ∪ ( 𝐹 “ 𝐴 ) ∈ ( ω ∪ ran ℵ ) ) |
20 |
18 19
|
syl6ibr |
⊢ ( 𝐴 ∈ V → ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) → ( card ‘ ∪ ( 𝐹 “ 𝐴 ) ) = ∪ ( 𝐹 “ 𝐴 ) ) ) |
21 |
20
|
adantrd |
⊢ ( 𝐴 ∈ V → ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ( 𝑥 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ ) ) → ( card ‘ ∪ ( 𝐹 “ 𝐴 ) ) = ∪ ( 𝐹 “ 𝐴 ) ) ) |
22 |
17 21
|
jcad |
⊢ ( 𝐴 ∈ V → ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ( 𝑥 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ ) ) → ( ω ⊆ ∪ ( 𝐹 “ 𝐴 ) ∧ ( card ‘ ∪ ( 𝐹 “ 𝐴 ) ) = ∪ ( 𝐹 “ 𝐴 ) ) ) ) |
23 |
|
isinfcard |
⊢ ( ( ω ⊆ ∪ ( 𝐹 “ 𝐴 ) ∧ ( card ‘ ∪ ( 𝐹 “ 𝐴 ) ) = ∪ ( 𝐹 “ 𝐴 ) ) ↔ ∪ ( 𝐹 “ 𝐴 ) ∈ ran ℵ ) |
24 |
22 23
|
syl6ib |
⊢ ( 𝐴 ∈ V → ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ( 𝑥 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ ) ) → ∪ ( 𝐹 “ 𝐴 ) ∈ ran ℵ ) ) |
25 |
24
|
exp4d |
⊢ ( 𝐴 ∈ V → ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) → ( 𝑥 ∈ 𝐴 → ( ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ → ∪ ( 𝐹 “ 𝐴 ) ∈ ran ℵ ) ) ) ) |
26 |
25
|
imp |
⊢ ( ( 𝐴 ∈ V ∧ 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ) → ( 𝑥 ∈ 𝐴 → ( ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ → ∪ ( 𝐹 “ 𝐴 ) ∈ ran ℵ ) ) ) |
27 |
26
|
rexlimdv |
⊢ ( ( 𝐴 ∈ V ∧ 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ) → ( ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ → ∪ ( 𝐹 “ 𝐴 ) ∈ ran ℵ ) ) |
28 |
27
|
expimpd |
⊢ ( 𝐴 ∈ V → ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ ) → ∪ ( 𝐹 “ 𝐴 ) ∈ ran ℵ ) ) |
29 |
1 28
|
syl |
⊢ ( 𝐴 ∈ 𝐵 → ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ ran ℵ ) → ∪ ( 𝐹 “ 𝐴 ) ∈ ran ℵ ) ) |