| Step |
Hyp |
Ref |
Expression |
| 1 |
|
eqid |
⊢ ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) = ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 2 |
1
|
elrnmpt |
⊢ ( 𝑧 ∈ V → ( 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ↔ ∃ 𝑦 ∈ 𝐵 𝑧 = ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 3 |
2
|
elv |
⊢ ( 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ↔ ∃ 𝑦 ∈ 𝐵 𝑧 = ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 4 |
|
simpr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = ( ◡ 𝐹 “ { 𝑦 } ) ) → 𝑧 = ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 5 |
|
simpl3 |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → 𝐵 ⊆ ran 𝐹 ) |
| 6 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
| 7 |
5 6
|
sseldd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ ran 𝐹 ) |
| 8 |
|
inisegn0 |
⊢ ( 𝑦 ∈ ran 𝐹 ↔ ( ◡ 𝐹 “ { 𝑦 } ) ≠ ∅ ) |
| 9 |
7 8
|
sylib |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → ( ◡ 𝐹 “ { 𝑦 } ) ≠ ∅ ) |
| 10 |
9
|
adantr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = ( ◡ 𝐹 “ { 𝑦 } ) ) → ( ◡ 𝐹 “ { 𝑦 } ) ≠ ∅ ) |
| 11 |
4 10
|
eqnetrd |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑧 = ( ◡ 𝐹 “ { 𝑦 } ) ) → 𝑧 ≠ ∅ ) |
| 12 |
11
|
r19.29an |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ ∃ 𝑦 ∈ 𝐵 𝑧 = ( ◡ 𝐹 “ { 𝑦 } ) ) → 𝑧 ≠ ∅ ) |
| 13 |
3 12
|
sylan2b |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) → 𝑧 ≠ ∅ ) |
| 14 |
13
|
ralrimiva |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) 𝑧 ≠ ∅ ) |
| 15 |
|
simp2 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → 𝐹 Fn 𝐴 ) |
| 16 |
|
simp1 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → 𝐴 ∈ 𝑉 ) |
| 17 |
15 16
|
jca |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ) ) |
| 18 |
|
fnex |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ) → 𝐹 ∈ V ) |
| 19 |
|
rnexg |
⊢ ( 𝐹 ∈ V → ran 𝐹 ∈ V ) |
| 20 |
17 18 19
|
3syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ran 𝐹 ∈ V ) |
| 21 |
|
simp3 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → 𝐵 ⊆ ran 𝐹 ) |
| 22 |
20 21
|
ssexd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → 𝐵 ∈ V ) |
| 23 |
|
mptexg |
⊢ ( 𝐵 ∈ V → ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∈ V ) |
| 24 |
|
rnexg |
⊢ ( ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∈ V → ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∈ V ) |
| 25 |
22 23 24
|
3syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∈ V ) |
| 26 |
|
fvi |
⊢ ( ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∈ V → ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) = ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 27 |
25 26
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) = ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 28 |
27
|
raleqdv |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( ∀ 𝑧 ∈ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) 𝑧 ≠ ∅ ↔ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) 𝑧 ≠ ∅ ) ) |
| 29 |
14 28
|
mpbird |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∀ 𝑧 ∈ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) 𝑧 ≠ ∅ ) |
| 30 |
|
fvex |
⊢ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∈ V |
| 31 |
30
|
ac5b |
⊢ ( ∀ 𝑧 ∈ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) 𝑧 ≠ ∅ → ∃ 𝑓 ( 𝑓 : ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ⟶ ∪ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) |
| 32 |
29 31
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∃ 𝑓 ( 𝑓 : ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ⟶ ∪ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) |
| 33 |
27
|
unieqd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∪ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) = ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 34 |
27 33
|
feq23d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( 𝑓 : ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ⟶ ∪ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ↔ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ) |
| 35 |
27
|
raleqdv |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( ∀ 𝑧 ∈ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ↔ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) |
| 36 |
34 35
|
anbi12d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( ( 𝑓 : ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ⟶ ∪ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ↔ ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) ) |
| 37 |
36
|
exbidv |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( ∃ 𝑓 ( 𝑓 : ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ⟶ ∪ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ↔ ∃ 𝑓 ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) ) |
| 38 |
32 37
|
mpbid |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∃ 𝑓 ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) |
| 39 |
|
vex |
⊢ 𝑓 ∈ V |
| 40 |
39
|
rnex |
⊢ ran 𝑓 ∈ V |
| 41 |
40
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ran 𝑓 ∈ V ) |
| 42 |
|
simplr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 43 |
|
frn |
⊢ ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) → ran 𝑓 ⊆ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 44 |
42 43
|
syl |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ran 𝑓 ⊆ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 45 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) |
| 46 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑓 |
| 47 |
|
nfmpt1 |
⊢ Ⅎ 𝑦 ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 48 |
47
|
nfrn |
⊢ Ⅎ 𝑦 ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 49 |
48
|
nfuni |
⊢ Ⅎ 𝑦 ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 50 |
46 48 49
|
nff |
⊢ Ⅎ 𝑦 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 51 |
45 50
|
nfan |
⊢ Ⅎ 𝑦 ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 52 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 |
| 53 |
48 52
|
nfralw |
⊢ Ⅎ 𝑦 ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 |
| 54 |
51 53
|
nfan |
⊢ Ⅎ 𝑦 ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) |
| 55 |
17 18
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → 𝐹 ∈ V ) |
| 56 |
55
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) → 𝐹 ∈ V ) |
| 57 |
|
cnvexg |
⊢ ( 𝐹 ∈ V → ◡ 𝐹 ∈ V ) |
| 58 |
|
imaexg |
⊢ ( ◡ 𝐹 ∈ V → ( ◡ 𝐹 “ { 𝑦 } ) ∈ V ) |
| 59 |
56 57 58
|
3syl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) → ( ◡ 𝐹 “ { 𝑦 } ) ∈ V ) |
| 60 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ { 𝑦 } ) ⊆ dom 𝐹 |
| 61 |
60
|
a1i |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) → ( ◡ 𝐹 “ { 𝑦 } ) ⊆ dom 𝐹 ) |
| 62 |
15
|
fndmd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → dom 𝐹 = 𝐴 ) |
| 63 |
62
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) → dom 𝐹 = 𝐴 ) |
| 64 |
61 63
|
sseqtrd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) → ( ◡ 𝐹 “ { 𝑦 } ) ⊆ 𝐴 ) |
| 65 |
59 64
|
elpwd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) → ( ◡ 𝐹 “ { 𝑦 } ) ∈ 𝒫 𝐴 ) |
| 66 |
65
|
ex |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑦 ∈ 𝐵 → ( ◡ 𝐹 “ { 𝑦 } ) ∈ 𝒫 𝐴 ) ) |
| 67 |
54 66
|
ralrimi |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ { 𝑦 } ) ∈ 𝒫 𝐴 ) |
| 68 |
1
|
rnmptss |
⊢ ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ { 𝑦 } ) ∈ 𝒫 𝐴 → ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⊆ 𝒫 𝐴 ) |
| 69 |
67 68
|
syl |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⊆ 𝒫 𝐴 ) |
| 70 |
|
sspwuni |
⊢ ( ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⊆ 𝒫 𝐴 ↔ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⊆ 𝐴 ) |
| 71 |
69 70
|
sylib |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⊆ 𝐴 ) |
| 72 |
44 71
|
sstrd |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ran 𝑓 ⊆ 𝐴 ) |
| 73 |
41 72
|
elpwd |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ran 𝑓 ∈ 𝒫 𝐴 ) |
| 74 |
|
fnfun |
⊢ ( 𝐹 Fn 𝐴 → Fun 𝐹 ) |
| 75 |
15 74
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → Fun 𝐹 ) |
| 76 |
75
|
ad5antr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) ) → Fun 𝐹 ) |
| 77 |
|
sndisj |
⊢ Disj 𝑦 ∈ 𝐵 { 𝑦 } |
| 78 |
|
disjpreima |
⊢ ( ( Fun 𝐹 ∧ Disj 𝑦 ∈ 𝐵 { 𝑦 } ) → Disj 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 79 |
76 77 78
|
sylancl |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) ) → Disj 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 80 |
|
disjrnmpt |
⊢ ( Disj 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ { 𝑦 } ) → Disj 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) 𝑧 ) |
| 81 |
79 80
|
syl |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) ) → Disj 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) 𝑧 ) |
| 82 |
|
simpllr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) ) → 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 83 |
|
simplr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) ) → 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 84 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) ) → ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) |
| 85 |
|
fveq2 |
⊢ ( 𝑧 = 𝑢 → ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑢 ) ) |
| 86 |
|
id |
⊢ ( 𝑧 = 𝑢 → 𝑧 = 𝑢 ) |
| 87 |
85 86
|
eleq12d |
⊢ ( 𝑧 = 𝑢 → ( ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ↔ ( 𝑓 ‘ 𝑢 ) ∈ 𝑢 ) ) |
| 88 |
87
|
rspcv |
⊢ ( 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) → ( ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 → ( 𝑓 ‘ 𝑢 ) ∈ 𝑢 ) ) |
| 89 |
88
|
imp |
⊢ ( ( 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑓 ‘ 𝑢 ) ∈ 𝑢 ) |
| 90 |
82 84 89
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) ) → ( 𝑓 ‘ 𝑢 ) ∈ 𝑢 ) |
| 91 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) ) → ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) ) |
| 92 |
|
fveq2 |
⊢ ( 𝑧 = 𝑣 → ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑣 ) ) |
| 93 |
|
id |
⊢ ( 𝑧 = 𝑣 → 𝑧 = 𝑣 ) |
| 94 |
92 93
|
eleq12d |
⊢ ( 𝑧 = 𝑣 → ( ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ↔ ( 𝑓 ‘ 𝑣 ) ∈ 𝑣 ) ) |
| 95 |
94
|
rspcv |
⊢ ( 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) → ( ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 → ( 𝑓 ‘ 𝑣 ) ∈ 𝑣 ) ) |
| 96 |
95
|
imp |
⊢ ( ( 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑓 ‘ 𝑣 ) ∈ 𝑣 ) |
| 97 |
83 84 96
|
syl2anc |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) ) → ( 𝑓 ‘ 𝑣 ) ∈ 𝑣 ) |
| 98 |
91 97
|
eqeltrd |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) ) → ( 𝑓 ‘ 𝑢 ) ∈ 𝑣 ) |
| 99 |
86 93
|
disji |
⊢ ( ( Disj 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) 𝑧 ∧ ( 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ( ( 𝑓 ‘ 𝑢 ) ∈ 𝑢 ∧ ( 𝑓 ‘ 𝑢 ) ∈ 𝑣 ) ) → 𝑢 = 𝑣 ) |
| 100 |
81 82 83 90 98 99
|
syl122anc |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) ) → 𝑢 = 𝑣 ) |
| 101 |
100
|
ex |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) → ( ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) → 𝑢 = 𝑣 ) ) |
| 102 |
101
|
anasss |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ ( 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ) → ( ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) → 𝑢 = 𝑣 ) ) |
| 103 |
102
|
ralrimivva |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ∀ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∀ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) → 𝑢 = 𝑣 ) ) |
| 104 |
42 103
|
jca |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∀ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) → 𝑢 = 𝑣 ) ) ) |
| 105 |
|
dff13 |
⊢ ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) –1-1→ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ↔ ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∀ 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( ( 𝑓 ‘ 𝑢 ) = ( 𝑓 ‘ 𝑣 ) → 𝑢 = 𝑣 ) ) ) |
| 106 |
104 105
|
sylibr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) –1-1→ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 107 |
|
f1f1orn |
⊢ ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) –1-1→ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) → 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) –1-1-onto→ ran 𝑓 ) |
| 108 |
106 107
|
syl |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) –1-1-onto→ ran 𝑓 ) |
| 109 |
|
f1oen3g |
⊢ ( ( 𝑓 ∈ V ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) –1-1-onto→ ran 𝑓 ) → ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ≈ ran 𝑓 ) |
| 110 |
39 108 109
|
sylancr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ≈ ran 𝑓 ) |
| 111 |
110
|
ensymd |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ran 𝑓 ≈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 112 |
22 23
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∈ V ) |
| 113 |
112
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∈ V ) |
| 114 |
59
|
ex |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑦 ∈ 𝐵 → ( ◡ 𝐹 “ { 𝑦 } ) ∈ V ) ) |
| 115 |
54 114
|
ralrimi |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ { 𝑦 } ) ∈ V ) |
| 116 |
75
|
ad5antr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑡 ∈ 𝐵 ) ∧ 𝑦 ≠ 𝑡 ) → Fun 𝐹 ) |
| 117 |
|
simpr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑡 ∈ 𝐵 ) ∧ 𝑦 ≠ 𝑡 ) → 𝑦 ≠ 𝑡 ) |
| 118 |
21
|
ad5antr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑡 ∈ 𝐵 ) ∧ 𝑦 ≠ 𝑡 ) → 𝐵 ⊆ ran 𝐹 ) |
| 119 |
|
simpllr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑡 ∈ 𝐵 ) ∧ 𝑦 ≠ 𝑡 ) → 𝑦 ∈ 𝐵 ) |
| 120 |
118 119
|
sseldd |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑡 ∈ 𝐵 ) ∧ 𝑦 ≠ 𝑡 ) → 𝑦 ∈ ran 𝐹 ) |
| 121 |
|
simplr |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑡 ∈ 𝐵 ) ∧ 𝑦 ≠ 𝑡 ) → 𝑡 ∈ 𝐵 ) |
| 122 |
118 121
|
sseldd |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑡 ∈ 𝐵 ) ∧ 𝑦 ≠ 𝑡 ) → 𝑡 ∈ ran 𝐹 ) |
| 123 |
116 117 120 122
|
preimane |
⊢ ( ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑡 ∈ 𝐵 ) ∧ 𝑦 ≠ 𝑡 ) → ( ◡ 𝐹 “ { 𝑦 } ) ≠ ( ◡ 𝐹 “ { 𝑡 } ) ) |
| 124 |
123
|
ex |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑡 ∈ 𝐵 ) → ( 𝑦 ≠ 𝑡 → ( ◡ 𝐹 “ { 𝑦 } ) ≠ ( ◡ 𝐹 “ { 𝑡 } ) ) ) |
| 125 |
124
|
necon4d |
⊢ ( ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑡 ∈ 𝐵 ) → ( ( ◡ 𝐹 “ { 𝑦 } ) = ( ◡ 𝐹 “ { 𝑡 } ) → 𝑦 = 𝑡 ) ) |
| 126 |
125
|
ralrimiva |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑦 ∈ 𝐵 ) → ∀ 𝑡 ∈ 𝐵 ( ( ◡ 𝐹 “ { 𝑦 } ) = ( ◡ 𝐹 “ { 𝑡 } ) → 𝑦 = 𝑡 ) ) |
| 127 |
126
|
ex |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑦 ∈ 𝐵 → ∀ 𝑡 ∈ 𝐵 ( ( ◡ 𝐹 “ { 𝑦 } ) = ( ◡ 𝐹 “ { 𝑡 } ) → 𝑦 = 𝑡 ) ) ) |
| 128 |
54 127
|
ralrimi |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ∀ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐵 ( ( ◡ 𝐹 “ { 𝑦 } ) = ( ◡ 𝐹 “ { 𝑡 } ) → 𝑦 = 𝑡 ) ) |
| 129 |
115 128
|
jca |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ { 𝑦 } ) ∈ V ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐵 ( ( ◡ 𝐹 “ { 𝑦 } ) = ( ◡ 𝐹 “ { 𝑡 } ) → 𝑦 = 𝑡 ) ) ) |
| 130 |
|
sneq |
⊢ ( 𝑦 = 𝑡 → { 𝑦 } = { 𝑡 } ) |
| 131 |
130
|
imaeq2d |
⊢ ( 𝑦 = 𝑡 → ( ◡ 𝐹 “ { 𝑦 } ) = ( ◡ 𝐹 “ { 𝑡 } ) ) |
| 132 |
1 131
|
f1mpt |
⊢ ( ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) : 𝐵 –1-1→ V ↔ ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ { 𝑦 } ) ∈ V ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐵 ( ( ◡ 𝐹 “ { 𝑦 } ) = ( ◡ 𝐹 “ { 𝑡 } ) → 𝑦 = 𝑡 ) ) ) |
| 133 |
129 132
|
sylibr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) : 𝐵 –1-1→ V ) |
| 134 |
|
f1f1orn |
⊢ ( ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) : 𝐵 –1-1→ V → ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) : 𝐵 –1-1-onto→ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 135 |
133 134
|
syl |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) : 𝐵 –1-1-onto→ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 136 |
|
f1oen3g |
⊢ ( ( ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∈ V ∧ ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) : 𝐵 –1-1-onto→ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) → 𝐵 ≈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 137 |
113 135 136
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → 𝐵 ≈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 138 |
137
|
ensymd |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ≈ 𝐵 ) |
| 139 |
|
entr |
⊢ ( ( ran 𝑓 ≈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ≈ 𝐵 ) → ran 𝑓 ≈ 𝐵 ) |
| 140 |
111 138 139
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ran 𝑓 ≈ 𝐵 ) |
| 141 |
|
imass2 |
⊢ ( ran 𝑓 ⊆ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) → ( 𝐹 “ ran 𝑓 ) ⊆ ( 𝐹 “ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ) |
| 142 |
43 141
|
syl |
⊢ ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) → ( 𝐹 “ ran 𝑓 ) ⊆ ( 𝐹 “ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ) |
| 143 |
42 142
|
syl |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝐹 “ ran 𝑓 ) ⊆ ( 𝐹 “ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ) |
| 144 |
|
imauni |
⊢ ( 𝐹 “ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) = ∪ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝐹 “ 𝑧 ) |
| 145 |
|
imaeq2 |
⊢ ( 𝑧 = ( ◡ 𝐹 “ { 𝑦 } ) → ( 𝐹 “ 𝑧 ) = ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 146 |
55
|
adantr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → 𝐹 ∈ V ) |
| 147 |
146 57 58
|
3syl |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → ( ◡ 𝐹 “ { 𝑦 } ) ∈ V ) |
| 148 |
145 147
|
iunrnmptss |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∪ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝐹 “ 𝑧 ) ⊆ ∪ 𝑦 ∈ 𝐵 ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 149 |
|
funimacnv |
⊢ ( Fun 𝐹 → ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) = ( { 𝑦 } ∩ ran 𝐹 ) ) |
| 150 |
75 149
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) = ( { 𝑦 } ∩ ran 𝐹 ) ) |
| 151 |
150
|
adantr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) = ( { 𝑦 } ∩ ran 𝐹 ) ) |
| 152 |
6
|
snssd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → { 𝑦 } ⊆ 𝐵 ) |
| 153 |
152 5
|
sstrd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → { 𝑦 } ⊆ ran 𝐹 ) |
| 154 |
|
df-ss |
⊢ ( { 𝑦 } ⊆ ran 𝐹 ↔ ( { 𝑦 } ∩ ran 𝐹 ) = { 𝑦 } ) |
| 155 |
153 154
|
sylib |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → ( { 𝑦 } ∩ ran 𝐹 ) = { 𝑦 } ) |
| 156 |
151 155
|
eqtrd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) = { 𝑦 } ) |
| 157 |
156
|
iuneq2dv |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∪ 𝑦 ∈ 𝐵 ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) = ∪ 𝑦 ∈ 𝐵 { 𝑦 } ) |
| 158 |
|
iunid |
⊢ ∪ 𝑦 ∈ 𝐵 { 𝑦 } = 𝐵 |
| 159 |
157 158
|
eqtrdi |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∪ 𝑦 ∈ 𝐵 ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) = 𝐵 ) |
| 160 |
148 159
|
sseqtrd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∪ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝐹 “ 𝑧 ) ⊆ 𝐵 ) |
| 161 |
160
|
ad2antrr |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ∪ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝐹 “ 𝑧 ) ⊆ 𝐵 ) |
| 162 |
144 161
|
eqsstrid |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝐹 “ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ⊆ 𝐵 ) |
| 163 |
143 162
|
sstrd |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝐹 “ ran 𝑓 ) ⊆ 𝐵 ) |
| 164 |
42
|
adantr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 165 |
164
|
ffund |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → Fun 𝑓 ) |
| 166 |
|
simpr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → 𝑡 ∈ 𝐵 ) |
| 167 |
55 57
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ◡ 𝐹 ∈ V ) |
| 168 |
167
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → ◡ 𝐹 ∈ V ) |
| 169 |
|
imaexg |
⊢ ( ◡ 𝐹 ∈ V → ( ◡ 𝐹 “ { 𝑡 } ) ∈ V ) |
| 170 |
168 169
|
syl |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → ( ◡ 𝐹 “ { 𝑡 } ) ∈ V ) |
| 171 |
1 131
|
elrnmpt1s |
⊢ ( ( 𝑡 ∈ 𝐵 ∧ ( ◡ 𝐹 “ { 𝑡 } ) ∈ V ) → ( ◡ 𝐹 “ { 𝑡 } ) ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 172 |
166 170 171
|
syl2anc |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → ( ◡ 𝐹 “ { 𝑡 } ) ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 173 |
164
|
fdmd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → dom 𝑓 = ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 174 |
172 173
|
eleqtrrd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → ( ◡ 𝐹 “ { 𝑡 } ) ∈ dom 𝑓 ) |
| 175 |
|
fvelrn |
⊢ ( ( Fun 𝑓 ∧ ( ◡ 𝐹 “ { 𝑡 } ) ∈ dom 𝑓 ) → ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ran 𝑓 ) |
| 176 |
165 174 175
|
syl2anc |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ran 𝑓 ) |
| 177 |
15
|
ad3antrrr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → 𝐹 Fn 𝐴 ) |
| 178 |
|
simplr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) |
| 179 |
|
fveq2 |
⊢ ( 𝑧 = ( ◡ 𝐹 “ { 𝑡 } ) → ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ) |
| 180 |
|
id |
⊢ ( 𝑧 = ( ◡ 𝐹 “ { 𝑡 } ) → 𝑧 = ( ◡ 𝐹 “ { 𝑡 } ) ) |
| 181 |
179 180
|
eleq12d |
⊢ ( 𝑧 = ( ◡ 𝐹 “ { 𝑡 } ) → ( ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ↔ ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ( ◡ 𝐹 “ { 𝑡 } ) ) ) |
| 182 |
181
|
rspcv |
⊢ ( ( ◡ 𝐹 “ { 𝑡 } ) ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) → ( ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 → ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ( ◡ 𝐹 “ { 𝑡 } ) ) ) |
| 183 |
182
|
imp |
⊢ ( ( ( ◡ 𝐹 “ { 𝑡 } ) ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ( ◡ 𝐹 “ { 𝑡 } ) ) |
| 184 |
172 178 183
|
syl2anc |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ( ◡ 𝐹 “ { 𝑡 } ) ) |
| 185 |
|
fniniseg |
⊢ ( 𝐹 Fn 𝐴 → ( ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ( ◡ 𝐹 “ { 𝑡 } ) ↔ ( ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ) = 𝑡 ) ) ) |
| 186 |
185
|
simplbda |
⊢ ( ( 𝐹 Fn 𝐴 ∧ ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ( ◡ 𝐹 “ { 𝑡 } ) ) → ( 𝐹 ‘ ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ) = 𝑡 ) |
| 187 |
177 184 186
|
syl2anc |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ) = 𝑡 ) |
| 188 |
|
fveqeq2 |
⊢ ( 𝑘 = ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) → ( ( 𝐹 ‘ 𝑘 ) = 𝑡 ↔ ( 𝐹 ‘ ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ) = 𝑡 ) ) |
| 189 |
188
|
rspcev |
⊢ ( ( ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ran 𝑓 ∧ ( 𝐹 ‘ ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ) = 𝑡 ) → ∃ 𝑘 ∈ ran 𝑓 ( 𝐹 ‘ 𝑘 ) = 𝑡 ) |
| 190 |
176 187 189
|
syl2anc |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → ∃ 𝑘 ∈ ran 𝑓 ( 𝐹 ‘ 𝑘 ) = 𝑡 ) |
| 191 |
72
|
adantr |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → ran 𝑓 ⊆ 𝐴 ) |
| 192 |
177 191
|
fvelimabd |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → ( 𝑡 ∈ ( 𝐹 “ ran 𝑓 ) ↔ ∃ 𝑘 ∈ ran 𝑓 ( 𝐹 ‘ 𝑘 ) = 𝑡 ) ) |
| 193 |
190 192
|
mpbird |
⊢ ( ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ∧ 𝑡 ∈ 𝐵 ) → 𝑡 ∈ ( 𝐹 “ ran 𝑓 ) ) |
| 194 |
193
|
ex |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑡 ∈ 𝐵 → 𝑡 ∈ ( 𝐹 “ ran 𝑓 ) ) ) |
| 195 |
194
|
ssrdv |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → 𝐵 ⊆ ( 𝐹 “ ran 𝑓 ) ) |
| 196 |
163 195
|
eqssd |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝐹 “ ran 𝑓 ) = 𝐵 ) |
| 197 |
140 196
|
jca |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( ran 𝑓 ≈ 𝐵 ∧ ( 𝐹 “ ran 𝑓 ) = 𝐵 ) ) |
| 198 |
|
breq1 |
⊢ ( 𝑥 = ran 𝑓 → ( 𝑥 ≈ 𝐵 ↔ ran 𝑓 ≈ 𝐵 ) ) |
| 199 |
|
imaeq2 |
⊢ ( 𝑥 = ran 𝑓 → ( 𝐹 “ 𝑥 ) = ( 𝐹 “ ran 𝑓 ) ) |
| 200 |
199
|
eqeq1d |
⊢ ( 𝑥 = ran 𝑓 → ( ( 𝐹 “ 𝑥 ) = 𝐵 ↔ ( 𝐹 “ ran 𝑓 ) = 𝐵 ) ) |
| 201 |
198 200
|
anbi12d |
⊢ ( 𝑥 = ran 𝑓 → ( ( 𝑥 ≈ 𝐵 ∧ ( 𝐹 “ 𝑥 ) = 𝐵 ) ↔ ( ran 𝑓 ≈ 𝐵 ∧ ( 𝐹 “ ran 𝑓 ) = 𝐵 ) ) ) |
| 202 |
201
|
rspcev |
⊢ ( ( ran 𝑓 ∈ 𝒫 𝐴 ∧ ( ran 𝑓 ≈ 𝐵 ∧ ( 𝐹 “ ran 𝑓 ) = 𝐵 ) ) → ∃ 𝑥 ∈ 𝒫 𝐴 ( 𝑥 ≈ 𝐵 ∧ ( 𝐹 “ 𝑥 ) = 𝐵 ) ) |
| 203 |
73 197 202
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ∃ 𝑥 ∈ 𝒫 𝐴 ( 𝑥 ≈ 𝐵 ∧ ( 𝐹 “ 𝑥 ) = 𝐵 ) ) |
| 204 |
203
|
anasss |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) → ∃ 𝑥 ∈ 𝒫 𝐴 ( 𝑥 ≈ 𝐵 ∧ ( 𝐹 “ 𝑥 ) = 𝐵 ) ) |
| 205 |
204
|
ex |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ∃ 𝑥 ∈ 𝒫 𝐴 ( 𝑥 ≈ 𝐵 ∧ ( 𝐹 “ 𝑥 ) = 𝐵 ) ) ) |
| 206 |
205
|
exlimdv |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( ∃ 𝑓 ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ∃ 𝑥 ∈ 𝒫 𝐴 ( 𝑥 ≈ 𝐵 ∧ ( 𝐹 “ 𝑥 ) = 𝐵 ) ) ) |
| 207 |
38 206
|
mpd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∃ 𝑥 ∈ 𝒫 𝐴 ( 𝑥 ≈ 𝐵 ∧ ( 𝐹 “ 𝑥 ) = 𝐵 ) ) |