| 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 |
|
fvi |
⊢ ( ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∈ V → ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) = ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 26 |
22 23 24 25
|
4syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) = ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 27 |
14 26
|
raleqtrrdv |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∀ 𝑧 ∈ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) 𝑧 ≠ ∅ ) |
| 28 |
|
fvex |
⊢ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ∈ V |
| 29 |
28
|
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 ) ) |
| 30 |
27 29
|
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 ) ) |
| 31 |
26
|
unieqd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∪ ( I ‘ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) = ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 32 |
26 31
|
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 } ) ) ) ) |
| 33 |
26
|
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 ) ) |
| 34 |
32 33
|
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 ) ) ) |
| 35 |
34
|
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 ) ) ) |
| 36 |
30 35
|
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 ) ) |
| 37 |
|
vex |
⊢ 𝑓 ∈ V |
| 38 |
37
|
rnex |
⊢ ran 𝑓 ∈ V |
| 39 |
38
|
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 ) |
| 40 |
|
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 } ) ) ) |
| 41 |
|
frn |
⊢ ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) → ran 𝑓 ⊆ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 42 |
40 41
|
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 } ) ) ) |
| 43 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) |
| 44 |
|
nfcv |
⊢ Ⅎ 𝑦 𝑓 |
| 45 |
|
nfmpt1 |
⊢ Ⅎ 𝑦 ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 46 |
45
|
nfrn |
⊢ Ⅎ 𝑦 ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 47 |
46
|
nfuni |
⊢ Ⅎ 𝑦 ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 48 |
44 46 47
|
nff |
⊢ Ⅎ 𝑦 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 49 |
43 48
|
nfan |
⊢ Ⅎ 𝑦 ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 50 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 |
| 51 |
46 50
|
nfralw |
⊢ Ⅎ 𝑦 ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 |
| 52 |
49 51
|
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 ) |
| 53 |
17 18
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → 𝐹 ∈ V ) |
| 54 |
53
|
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 ) |
| 55 |
|
cnvexg |
⊢ ( 𝐹 ∈ V → ◡ 𝐹 ∈ V ) |
| 56 |
|
imaexg |
⊢ ( ◡ 𝐹 ∈ V → ( ◡ 𝐹 “ { 𝑦 } ) ∈ V ) |
| 57 |
54 55 56
|
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 ) |
| 58 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ { 𝑦 } ) ⊆ dom 𝐹 |
| 59 |
58
|
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 ) |
| 60 |
15
|
fndmd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → dom 𝐹 = 𝐴 ) |
| 61 |
60
|
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 ) |
| 62 |
59 61
|
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 ) |
| 63 |
57 62
|
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 ) |
| 64 |
63
|
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 ) ) |
| 65 |
52 64
|
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 ) |
| 66 |
1
|
rnmptss |
⊢ ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ { 𝑦 } ) ∈ 𝒫 𝐴 → ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⊆ 𝒫 𝐴 ) |
| 67 |
65 66
|
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 ) |
| 68 |
|
sspwuni |
⊢ ( ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⊆ 𝒫 𝐴 ↔ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⊆ 𝐴 ) |
| 69 |
67 68
|
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 ) |
| 70 |
42 69
|
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 ) |
| 71 |
39 70
|
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 ) |
| 72 |
|
fnfun |
⊢ ( 𝐹 Fn 𝐴 → Fun 𝐹 ) |
| 73 |
15 72
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → Fun 𝐹 ) |
| 74 |
73
|
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 ) |
| 75 |
|
sndisj |
⊢ Disj 𝑦 ∈ 𝐵 { 𝑦 } |
| 76 |
|
disjpreima |
⊢ ( ( Fun 𝐹 ∧ Disj 𝑦 ∈ 𝐵 { 𝑦 } ) → Disj 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ { 𝑦 } ) ) |
| 77 |
74 75 76
|
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 } ) ) |
| 78 |
|
disjrnmpt |
⊢ ( Disj 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ { 𝑦 } ) → Disj 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) 𝑧 ) |
| 79 |
77 78
|
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 ) |
| 80 |
|
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 } ) ) ) |
| 81 |
|
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 } ) ) ) |
| 82 |
|
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 ) |
| 83 |
|
fveq2 |
⊢ ( 𝑧 = 𝑢 → ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑢 ) ) |
| 84 |
|
id |
⊢ ( 𝑧 = 𝑢 → 𝑧 = 𝑢 ) |
| 85 |
83 84
|
eleq12d |
⊢ ( 𝑧 = 𝑢 → ( ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ↔ ( 𝑓 ‘ 𝑢 ) ∈ 𝑢 ) ) |
| 86 |
85
|
rspcv |
⊢ ( 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) → ( ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 → ( 𝑓 ‘ 𝑢 ) ∈ 𝑢 ) ) |
| 87 |
86
|
imp |
⊢ ( ( 𝑢 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑓 ‘ 𝑢 ) ∈ 𝑢 ) |
| 88 |
80 82 87
|
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 ) |
| 89 |
|
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 ) ) |
| 90 |
|
fveq2 |
⊢ ( 𝑧 = 𝑣 → ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ 𝑣 ) ) |
| 91 |
|
id |
⊢ ( 𝑧 = 𝑣 → 𝑧 = 𝑣 ) |
| 92 |
90 91
|
eleq12d |
⊢ ( 𝑧 = 𝑣 → ( ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ↔ ( 𝑓 ‘ 𝑣 ) ∈ 𝑣 ) ) |
| 93 |
92
|
rspcv |
⊢ ( 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) → ( ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 → ( 𝑓 ‘ 𝑣 ) ∈ 𝑣 ) ) |
| 94 |
93
|
imp |
⊢ ( ( 𝑣 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑓 ‘ 𝑣 ) ∈ 𝑣 ) |
| 95 |
81 82 94
|
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 ) |
| 96 |
89 95
|
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 ) |
| 97 |
84 91
|
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 ) |
| 98 |
79 80 81 88 96 97
|
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 ) |
| 99 |
98
|
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 ) ) |
| 100 |
99
|
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 ) ) |
| 101 |
100
|
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 ) ) |
| 102 |
40 101
|
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 ) ) ) |
| 103 |
|
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 ) ) ) |
| 104 |
102 103
|
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 } ) ) ) |
| 105 |
|
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 ) |
| 106 |
104 105
|
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 ) |
| 107 |
|
f1oen3g |
⊢ ( ( 𝑓 ∈ V ∧ 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) –1-1-onto→ ran 𝑓 ) → ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ≈ ran 𝑓 ) |
| 108 |
37 106 107
|
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 ) |
| 109 |
108
|
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 } ) ) ) |
| 110 |
22 23
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∈ V ) |
| 111 |
110
|
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 ) |
| 112 |
57
|
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 ) ) |
| 113 |
52 112
|
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 ) |
| 114 |
73
|
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 ) |
| 115 |
|
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 ) |
| 116 |
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 ) |
| 117 |
|
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 ) |
| 118 |
116 117
|
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 ) |
| 119 |
|
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 ) |
| 120 |
116 119
|
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 ) → 𝑡 ∈ ran 𝐹 ) |
| 121 |
114 115 118 120
|
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 } ) ) |
| 122 |
121
|
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 } ) ) ) |
| 123 |
122
|
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 ) ) |
| 124 |
123
|
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 ) ) |
| 125 |
124
|
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 ) ) ) |
| 126 |
52 125
|
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 ) ) |
| 127 |
113 126
|
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 ) ) ) |
| 128 |
|
sneq |
⊢ ( 𝑦 = 𝑡 → { 𝑦 } = { 𝑡 } ) |
| 129 |
128
|
imaeq2d |
⊢ ( 𝑦 = 𝑡 → ( ◡ 𝐹 “ { 𝑦 } ) = ( ◡ 𝐹 “ { 𝑡 } ) ) |
| 130 |
1 129
|
f1mpt |
⊢ ( ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) : 𝐵 –1-1→ V ↔ ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ { 𝑦 } ) ∈ V ∧ ∀ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐵 ( ( ◡ 𝐹 “ { 𝑦 } ) = ( ◡ 𝐹 “ { 𝑡 } ) → 𝑦 = 𝑡 ) ) ) |
| 131 |
127 130
|
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 ) |
| 132 |
|
f1f1orn |
⊢ ( ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) : 𝐵 –1-1→ V → ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) : 𝐵 –1-1-onto→ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 133 |
131 132
|
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 } ) ) ) |
| 134 |
|
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 } ) ) ) |
| 135 |
111 133 134
|
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 } ) ) ) |
| 136 |
135
|
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 ) |
| 137 |
|
entr |
⊢ ( ( ran 𝑓 ≈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ≈ 𝐵 ) → ran 𝑓 ≈ 𝐵 ) |
| 138 |
109 136 137
|
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 ) |
| 139 |
|
imass2 |
⊢ ( ran 𝑓 ⊆ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) → ( 𝐹 “ ran 𝑓 ) ⊆ ( 𝐹 “ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ) |
| 140 |
41 139
|
syl |
⊢ ( 𝑓 : ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ⟶ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) → ( 𝐹 “ ran 𝑓 ) ⊆ ( 𝐹 “ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) ) |
| 141 |
40 140
|
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 } ) ) ) ) |
| 142 |
|
imauni |
⊢ ( 𝐹 “ ∪ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) = ∪ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝐹 “ 𝑧 ) |
| 143 |
|
imaeq2 |
⊢ ( 𝑧 = ( ◡ 𝐹 “ { 𝑦 } ) → ( 𝐹 “ 𝑧 ) = ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 144 |
53
|
adantr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → 𝐹 ∈ V ) |
| 145 |
144 55 56
|
3syl |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → ( ◡ 𝐹 “ { 𝑦 } ) ∈ V ) |
| 146 |
143 145
|
iunrnmptss |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∪ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝐹 “ 𝑧 ) ⊆ ∪ 𝑦 ∈ 𝐵 ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 147 |
|
funimacnv |
⊢ ( Fun 𝐹 → ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) = ( { 𝑦 } ∩ ran 𝐹 ) ) |
| 148 |
73 147
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) = ( { 𝑦 } ∩ ran 𝐹 ) ) |
| 149 |
148
|
adantr |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) = ( { 𝑦 } ∩ ran 𝐹 ) ) |
| 150 |
6
|
snssd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → { 𝑦 } ⊆ 𝐵 ) |
| 151 |
150 5
|
sstrd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → { 𝑦 } ⊆ ran 𝐹 ) |
| 152 |
|
dfss2 |
⊢ ( { 𝑦 } ⊆ ran 𝐹 ↔ ( { 𝑦 } ∩ ran 𝐹 ) = { 𝑦 } ) |
| 153 |
151 152
|
sylib |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → ( { 𝑦 } ∩ ran 𝐹 ) = { 𝑦 } ) |
| 154 |
149 153
|
eqtrd |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) = { 𝑦 } ) |
| 155 |
154
|
iuneq2dv |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∪ 𝑦 ∈ 𝐵 ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) = ∪ 𝑦 ∈ 𝐵 { 𝑦 } ) |
| 156 |
|
iunid |
⊢ ∪ 𝑦 ∈ 𝐵 { 𝑦 } = 𝐵 |
| 157 |
155 156
|
eqtrdi |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∪ 𝑦 ∈ 𝐵 ( 𝐹 “ ( ◡ 𝐹 “ { 𝑦 } ) ) = 𝐵 ) |
| 158 |
146 157
|
sseqtrd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∪ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝐹 “ 𝑧 ) ⊆ 𝐵 ) |
| 159 |
158
|
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 ) |
| 160 |
142 159
|
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 ) |
| 161 |
141 160
|
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 ) |
| 162 |
40
|
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 } ) ) ) |
| 163 |
162
|
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 ) |
| 164 |
|
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 ) |
| 165 |
53 55
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ◡ 𝐹 ∈ V ) |
| 166 |
165
|
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 ) |
| 167 |
|
imaexg |
⊢ ( ◡ 𝐹 ∈ V → ( ◡ 𝐹 “ { 𝑡 } ) ∈ V ) |
| 168 |
166 167
|
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 ) |
| 169 |
1 129
|
elrnmpt1s |
⊢ ( ( 𝑡 ∈ 𝐵 ∧ ( ◡ 𝐹 “ { 𝑡 } ) ∈ V ) → ( ◡ 𝐹 “ { 𝑡 } ) ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ) |
| 170 |
164 168 169
|
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 } ) ) ) |
| 171 |
162
|
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 } ) ) ) |
| 172 |
170 171
|
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 ) |
| 173 |
|
fvelrn |
⊢ ( ( Fun 𝑓 ∧ ( ◡ 𝐹 “ { 𝑡 } ) ∈ dom 𝑓 ) → ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ran 𝑓 ) |
| 174 |
163 172 173
|
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 ) |
| 175 |
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 ) |
| 176 |
|
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 ) |
| 177 |
|
fveq2 |
⊢ ( 𝑧 = ( ◡ 𝐹 “ { 𝑡 } ) → ( 𝑓 ‘ 𝑧 ) = ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ) |
| 178 |
|
id |
⊢ ( 𝑧 = ( ◡ 𝐹 “ { 𝑡 } ) → 𝑧 = ( ◡ 𝐹 “ { 𝑡 } ) ) |
| 179 |
177 178
|
eleq12d |
⊢ ( 𝑧 = ( ◡ 𝐹 “ { 𝑡 } ) → ( ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ↔ ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ( ◡ 𝐹 “ { 𝑡 } ) ) ) |
| 180 |
179
|
rspcv |
⊢ ( ( ◡ 𝐹 “ { 𝑡 } ) ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) → ( ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 → ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ( ◡ 𝐹 “ { 𝑡 } ) ) ) |
| 181 |
180
|
imp |
⊢ ( ( ( ◡ 𝐹 “ { 𝑡 } ) ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ∧ ∀ 𝑧 ∈ ran ( 𝑦 ∈ 𝐵 ↦ ( ◡ 𝐹 “ { 𝑦 } ) ) ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ( ◡ 𝐹 “ { 𝑡 } ) ) |
| 182 |
170 176 181
|
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 } ) ) |
| 183 |
|
fniniseg |
⊢ ( 𝐹 Fn 𝐴 → ( ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ( ◡ 𝐹 “ { 𝑡 } ) ↔ ( ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ) = 𝑡 ) ) ) |
| 184 |
183
|
simplbda |
⊢ ( ( 𝐹 Fn 𝐴 ∧ ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ( ◡ 𝐹 “ { 𝑡 } ) ) → ( 𝐹 ‘ ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ) = 𝑡 ) |
| 185 |
175 182 184
|
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 ) |
| 186 |
|
fveqeq2 |
⊢ ( 𝑘 = ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) → ( ( 𝐹 ‘ 𝑘 ) = 𝑡 ↔ ( 𝐹 ‘ ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ) = 𝑡 ) ) |
| 187 |
186
|
rspcev |
⊢ ( ( ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ∈ ran 𝑓 ∧ ( 𝐹 ‘ ( 𝑓 ‘ ( ◡ 𝐹 “ { 𝑡 } ) ) ) = 𝑡 ) → ∃ 𝑘 ∈ ran 𝑓 ( 𝐹 ‘ 𝑘 ) = 𝑡 ) |
| 188 |
174 185 187
|
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 ) |
| 189 |
70
|
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 ) |
| 190 |
175 189
|
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 ) ) |
| 191 |
188 190
|
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 ) ) |
| 192 |
191
|
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 ) ) ) |
| 193 |
192
|
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 ) ) |
| 194 |
161 193
|
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 ) |
| 195 |
138 194
|
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 ) ) |
| 196 |
|
breq1 |
⊢ ( 𝑥 = ran 𝑓 → ( 𝑥 ≈ 𝐵 ↔ ran 𝑓 ≈ 𝐵 ) ) |
| 197 |
|
imaeq2 |
⊢ ( 𝑥 = ran 𝑓 → ( 𝐹 “ 𝑥 ) = ( 𝐹 “ ran 𝑓 ) ) |
| 198 |
197
|
eqeq1d |
⊢ ( 𝑥 = ran 𝑓 → ( ( 𝐹 “ 𝑥 ) = 𝐵 ↔ ( 𝐹 “ ran 𝑓 ) = 𝐵 ) ) |
| 199 |
196 198
|
anbi12d |
⊢ ( 𝑥 = ran 𝑓 → ( ( 𝑥 ≈ 𝐵 ∧ ( 𝐹 “ 𝑥 ) = 𝐵 ) ↔ ( ran 𝑓 ≈ 𝐵 ∧ ( 𝐹 “ ran 𝑓 ) = 𝐵 ) ) ) |
| 200 |
199
|
rspcev |
⊢ ( ( ran 𝑓 ∈ 𝒫 𝐴 ∧ ( ran 𝑓 ≈ 𝐵 ∧ ( 𝐹 “ ran 𝑓 ) = 𝐵 ) ) → ∃ 𝑥 ∈ 𝒫 𝐴 ( 𝑥 ≈ 𝐵 ∧ ( 𝐹 “ 𝑥 ) = 𝐵 ) ) |
| 201 |
71 195 200
|
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 ) ) |
| 202 |
201
|
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 ) ) |
| 203 |
202
|
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 ) ) ) |
| 204 |
203
|
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 ) ) ) |
| 205 |
36 204
|
mpd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹 ) → ∃ 𝑥 ∈ 𝒫 𝐴 ( 𝑥 ≈ 𝐵 ∧ ( 𝐹 “ 𝑥 ) = 𝐵 ) ) |