| Step |
Hyp |
Ref |
Expression |
| 1 |
|
filtop |
⊢ ( 𝐿 ∈ ( Fil ‘ 𝑋 ) → 𝑋 ∈ 𝐿 ) |
| 2 |
1
|
3ad2ant2 |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → 𝑋 ∈ 𝐿 ) |
| 3 |
|
simp1 |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → 𝑌 ∈ 𝐴 ) |
| 4 |
|
simp3 |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → 𝐹 : 𝑌 ⟶ 𝑋 ) |
| 5 |
|
fmf |
⊢ ( ( 𝑋 ∈ 𝐿 ∧ 𝑌 ∈ 𝐴 ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝑋 FilMap 𝐹 ) : ( fBas ‘ 𝑌 ) ⟶ ( Fil ‘ 𝑋 ) ) |
| 6 |
2 3 4 5
|
syl3anc |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝑋 FilMap 𝐹 ) : ( fBas ‘ 𝑌 ) ⟶ ( Fil ‘ 𝑋 ) ) |
| 7 |
6
|
ffnd |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝑋 FilMap 𝐹 ) Fn ( fBas ‘ 𝑌 ) ) |
| 8 |
|
fvelrnb |
⊢ ( ( 𝑋 FilMap 𝐹 ) Fn ( fBas ‘ 𝑌 ) → ( 𝐿 ∈ ran ( 𝑋 FilMap 𝐹 ) ↔ ∃ 𝑏 ∈ ( fBas ‘ 𝑌 ) ( ( 𝑋 FilMap 𝐹 ) ‘ 𝑏 ) = 𝐿 ) ) |
| 9 |
7 8
|
syl |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝐿 ∈ ran ( 𝑋 FilMap 𝐹 ) ↔ ∃ 𝑏 ∈ ( fBas ‘ 𝑌 ) ( ( 𝑋 FilMap 𝐹 ) ‘ 𝑏 ) = 𝐿 ) ) |
| 10 |
|
ffn |
⊢ ( 𝐹 : 𝑌 ⟶ 𝑋 → 𝐹 Fn 𝑌 ) |
| 11 |
|
dffn4 |
⊢ ( 𝐹 Fn 𝑌 ↔ 𝐹 : 𝑌 –onto→ ran 𝐹 ) |
| 12 |
10 11
|
sylib |
⊢ ( 𝐹 : 𝑌 ⟶ 𝑋 → 𝐹 : 𝑌 –onto→ ran 𝐹 ) |
| 13 |
|
foima |
⊢ ( 𝐹 : 𝑌 –onto→ ran 𝐹 → ( 𝐹 “ 𝑌 ) = ran 𝐹 ) |
| 14 |
12 13
|
syl |
⊢ ( 𝐹 : 𝑌 ⟶ 𝑋 → ( 𝐹 “ 𝑌 ) = ran 𝐹 ) |
| 15 |
14
|
ad2antlr |
⊢ ( ( ( 𝑋 ∈ 𝐿 ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝑏 ∈ ( fBas ‘ 𝑌 ) ) → ( 𝐹 “ 𝑌 ) = ran 𝐹 ) |
| 16 |
|
simpll |
⊢ ( ( ( 𝑋 ∈ 𝐿 ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝑏 ∈ ( fBas ‘ 𝑌 ) ) → 𝑋 ∈ 𝐿 ) |
| 17 |
|
simpr |
⊢ ( ( ( 𝑋 ∈ 𝐿 ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝑏 ∈ ( fBas ‘ 𝑌 ) ) → 𝑏 ∈ ( fBas ‘ 𝑌 ) ) |
| 18 |
|
simplr |
⊢ ( ( ( 𝑋 ∈ 𝐿 ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝑏 ∈ ( fBas ‘ 𝑌 ) ) → 𝐹 : 𝑌 ⟶ 𝑋 ) |
| 19 |
|
fgcl |
⊢ ( 𝑏 ∈ ( fBas ‘ 𝑌 ) → ( 𝑌 filGen 𝑏 ) ∈ ( Fil ‘ 𝑌 ) ) |
| 20 |
|
filtop |
⊢ ( ( 𝑌 filGen 𝑏 ) ∈ ( Fil ‘ 𝑌 ) → 𝑌 ∈ ( 𝑌 filGen 𝑏 ) ) |
| 21 |
19 20
|
syl |
⊢ ( 𝑏 ∈ ( fBas ‘ 𝑌 ) → 𝑌 ∈ ( 𝑌 filGen 𝑏 ) ) |
| 22 |
21
|
adantl |
⊢ ( ( ( 𝑋 ∈ 𝐿 ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝑏 ∈ ( fBas ‘ 𝑌 ) ) → 𝑌 ∈ ( 𝑌 filGen 𝑏 ) ) |
| 23 |
|
eqid |
⊢ ( 𝑌 filGen 𝑏 ) = ( 𝑌 filGen 𝑏 ) |
| 24 |
23
|
imaelfm |
⊢ ( ( ( 𝑋 ∈ 𝐿 ∧ 𝑏 ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝑌 ∈ ( 𝑌 filGen 𝑏 ) ) → ( 𝐹 “ 𝑌 ) ∈ ( ( 𝑋 FilMap 𝐹 ) ‘ 𝑏 ) ) |
| 25 |
16 17 18 22 24
|
syl31anc |
⊢ ( ( ( 𝑋 ∈ 𝐿 ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝑏 ∈ ( fBas ‘ 𝑌 ) ) → ( 𝐹 “ 𝑌 ) ∈ ( ( 𝑋 FilMap 𝐹 ) ‘ 𝑏 ) ) |
| 26 |
15 25
|
eqeltrrd |
⊢ ( ( ( 𝑋 ∈ 𝐿 ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝑏 ∈ ( fBas ‘ 𝑌 ) ) → ran 𝐹 ∈ ( ( 𝑋 FilMap 𝐹 ) ‘ 𝑏 ) ) |
| 27 |
|
eleq2 |
⊢ ( ( ( 𝑋 FilMap 𝐹 ) ‘ 𝑏 ) = 𝐿 → ( ran 𝐹 ∈ ( ( 𝑋 FilMap 𝐹 ) ‘ 𝑏 ) ↔ ran 𝐹 ∈ 𝐿 ) ) |
| 28 |
26 27
|
syl5ibcom |
⊢ ( ( ( 𝑋 ∈ 𝐿 ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ 𝑏 ∈ ( fBas ‘ 𝑌 ) ) → ( ( ( 𝑋 FilMap 𝐹 ) ‘ 𝑏 ) = 𝐿 → ran 𝐹 ∈ 𝐿 ) ) |
| 29 |
28
|
ex |
⊢ ( ( 𝑋 ∈ 𝐿 ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝑏 ∈ ( fBas ‘ 𝑌 ) → ( ( ( 𝑋 FilMap 𝐹 ) ‘ 𝑏 ) = 𝐿 → ran 𝐹 ∈ 𝐿 ) ) ) |
| 30 |
1 29
|
sylan |
⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝑏 ∈ ( fBas ‘ 𝑌 ) → ( ( ( 𝑋 FilMap 𝐹 ) ‘ 𝑏 ) = 𝐿 → ran 𝐹 ∈ 𝐿 ) ) ) |
| 31 |
30
|
3adant1 |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝑏 ∈ ( fBas ‘ 𝑌 ) → ( ( ( 𝑋 FilMap 𝐹 ) ‘ 𝑏 ) = 𝐿 → ran 𝐹 ∈ 𝐿 ) ) ) |
| 32 |
31
|
rexlimdv |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( ∃ 𝑏 ∈ ( fBas ‘ 𝑌 ) ( ( 𝑋 FilMap 𝐹 ) ‘ 𝑏 ) = 𝐿 → ran 𝐹 ∈ 𝐿 ) ) |
| 33 |
9 32
|
sylbid |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝐿 ∈ ran ( 𝑋 FilMap 𝐹 ) → ran 𝐹 ∈ 𝐿 ) ) |
| 34 |
|
simpl2 |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → 𝐿 ∈ ( Fil ‘ 𝑋 ) ) |
| 35 |
|
filelss |
⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐿 ) → 𝑡 ⊆ 𝑋 ) |
| 36 |
35
|
ex |
⊢ ( 𝐿 ∈ ( Fil ‘ 𝑋 ) → ( 𝑡 ∈ 𝐿 → 𝑡 ⊆ 𝑋 ) ) |
| 37 |
34 36
|
syl |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( 𝑡 ∈ 𝐿 → 𝑡 ⊆ 𝑋 ) ) |
| 38 |
|
simpr |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑡 ∈ 𝐿 ) → 𝑡 ∈ 𝐿 ) |
| 39 |
|
eqidd |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑡 ∈ 𝐿 ) → ( ◡ 𝐹 “ 𝑡 ) = ( ◡ 𝐹 “ 𝑡 ) ) |
| 40 |
|
imaeq2 |
⊢ ( 𝑥 = 𝑡 → ( ◡ 𝐹 “ 𝑥 ) = ( ◡ 𝐹 “ 𝑡 ) ) |
| 41 |
40
|
rspceeqv |
⊢ ( ( 𝑡 ∈ 𝐿 ∧ ( ◡ 𝐹 “ 𝑡 ) = ( ◡ 𝐹 “ 𝑡 ) ) → ∃ 𝑥 ∈ 𝐿 ( ◡ 𝐹 “ 𝑡 ) = ( ◡ 𝐹 “ 𝑥 ) ) |
| 42 |
38 39 41
|
syl2anc |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑡 ∈ 𝐿 ) → ∃ 𝑥 ∈ 𝐿 ( ◡ 𝐹 “ 𝑡 ) = ( ◡ 𝐹 “ 𝑥 ) ) |
| 43 |
|
simpl1 |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → 𝑌 ∈ 𝐴 ) |
| 44 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ 𝑡 ) ⊆ dom 𝐹 |
| 45 |
|
fdm |
⊢ ( 𝐹 : 𝑌 ⟶ 𝑋 → dom 𝐹 = 𝑌 ) |
| 46 |
44 45
|
sseqtrid |
⊢ ( 𝐹 : 𝑌 ⟶ 𝑋 → ( ◡ 𝐹 “ 𝑡 ) ⊆ 𝑌 ) |
| 47 |
46
|
3ad2ant3 |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( ◡ 𝐹 “ 𝑡 ) ⊆ 𝑌 ) |
| 48 |
47
|
adantr |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( ◡ 𝐹 “ 𝑡 ) ⊆ 𝑌 ) |
| 49 |
43 48
|
ssexd |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( ◡ 𝐹 “ 𝑡 ) ∈ V ) |
| 50 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) = ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) |
| 51 |
50
|
elrnmpt |
⊢ ( ( ◡ 𝐹 “ 𝑡 ) ∈ V → ( ( ◡ 𝐹 “ 𝑡 ) ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ↔ ∃ 𝑥 ∈ 𝐿 ( ◡ 𝐹 “ 𝑡 ) = ( ◡ 𝐹 “ 𝑥 ) ) ) |
| 52 |
49 51
|
syl |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( ( ◡ 𝐹 “ 𝑡 ) ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ↔ ∃ 𝑥 ∈ 𝐿 ( ◡ 𝐹 “ 𝑡 ) = ( ◡ 𝐹 “ 𝑥 ) ) ) |
| 53 |
52
|
adantr |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑡 ∈ 𝐿 ) → ( ( ◡ 𝐹 “ 𝑡 ) ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ↔ ∃ 𝑥 ∈ 𝐿 ( ◡ 𝐹 “ 𝑡 ) = ( ◡ 𝐹 “ 𝑥 ) ) ) |
| 54 |
42 53
|
mpbird |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑡 ∈ 𝐿 ) → ( ◡ 𝐹 “ 𝑡 ) ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ) |
| 55 |
|
ssid |
⊢ ( ◡ 𝐹 “ 𝑡 ) ⊆ ( ◡ 𝐹 “ 𝑡 ) |
| 56 |
|
ffun |
⊢ ( 𝐹 : 𝑌 ⟶ 𝑋 → Fun 𝐹 ) |
| 57 |
56
|
3ad2ant3 |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → Fun 𝐹 ) |
| 58 |
57
|
ad2antrr |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑡 ∈ 𝐿 ) → Fun 𝐹 ) |
| 59 |
|
funimass3 |
⊢ ( ( Fun 𝐹 ∧ ( ◡ 𝐹 “ 𝑡 ) ⊆ dom 𝐹 ) → ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑡 ) ) ⊆ 𝑡 ↔ ( ◡ 𝐹 “ 𝑡 ) ⊆ ( ◡ 𝐹 “ 𝑡 ) ) ) |
| 60 |
58 44 59
|
sylancl |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑡 ∈ 𝐿 ) → ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑡 ) ) ⊆ 𝑡 ↔ ( ◡ 𝐹 “ 𝑡 ) ⊆ ( ◡ 𝐹 “ 𝑡 ) ) ) |
| 61 |
55 60
|
mpbiri |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑡 ∈ 𝐿 ) → ( 𝐹 “ ( ◡ 𝐹 “ 𝑡 ) ) ⊆ 𝑡 ) |
| 62 |
|
imaeq2 |
⊢ ( 𝑠 = ( ◡ 𝐹 “ 𝑡 ) → ( 𝐹 “ 𝑠 ) = ( 𝐹 “ ( ◡ 𝐹 “ 𝑡 ) ) ) |
| 63 |
62
|
sseq1d |
⊢ ( 𝑠 = ( ◡ 𝐹 “ 𝑡 ) → ( ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ↔ ( 𝐹 “ ( ◡ 𝐹 “ 𝑡 ) ) ⊆ 𝑡 ) ) |
| 64 |
63
|
rspcev |
⊢ ( ( ( ◡ 𝐹 “ 𝑡 ) ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ∧ ( 𝐹 “ ( ◡ 𝐹 “ 𝑡 ) ) ⊆ 𝑡 ) → ∃ 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) |
| 65 |
54 61 64
|
syl2anc |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑡 ∈ 𝐿 ) → ∃ 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) |
| 66 |
65
|
ex |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( 𝑡 ∈ 𝐿 → ∃ 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) ) |
| 67 |
37 66
|
jcad |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( 𝑡 ∈ 𝐿 → ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) ) ) |
| 68 |
34
|
adantr |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ ( ( 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ∧ ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) ∧ 𝑡 ⊆ 𝑋 ) ) → 𝐿 ∈ ( Fil ‘ 𝑋 ) ) |
| 69 |
50
|
elrnmpt |
⊢ ( 𝑠 ∈ V → ( 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ↔ ∃ 𝑥 ∈ 𝐿 𝑠 = ( ◡ 𝐹 “ 𝑥 ) ) ) |
| 70 |
69
|
elv |
⊢ ( 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ↔ ∃ 𝑥 ∈ 𝐿 𝑠 = ( ◡ 𝐹 “ 𝑥 ) ) |
| 71 |
|
ssid |
⊢ ( ◡ 𝐹 “ 𝑥 ) ⊆ ( ◡ 𝐹 “ 𝑥 ) |
| 72 |
57
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → Fun 𝐹 ) |
| 73 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ 𝑥 ) ⊆ dom 𝐹 |
| 74 |
|
funimass3 |
⊢ ( ( Fun 𝐹 ∧ ( ◡ 𝐹 “ 𝑥 ) ⊆ dom 𝐹 ) → ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑥 ↔ ( ◡ 𝐹 “ 𝑥 ) ⊆ ( ◡ 𝐹 “ 𝑥 ) ) ) |
| 75 |
72 73 74
|
sylancl |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑥 ↔ ( ◡ 𝐹 “ 𝑥 ) ⊆ ( ◡ 𝐹 “ 𝑥 ) ) ) |
| 76 |
71 75
|
mpbiri |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑥 ) |
| 77 |
|
imassrn |
⊢ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ ran 𝐹 |
| 78 |
|
ssin |
⊢ ( ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑥 ∧ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ ran 𝐹 ) ↔ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ ( 𝑥 ∩ ran 𝐹 ) ) |
| 79 |
76 77 78
|
sylanblc |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ ( 𝑥 ∩ ran 𝐹 ) ) |
| 80 |
|
elin |
⊢ ( 𝑧 ∈ ( 𝑥 ∩ ran 𝐹 ) ↔ ( 𝑧 ∈ 𝑥 ∧ 𝑧 ∈ ran 𝐹 ) ) |
| 81 |
|
fvelrnb |
⊢ ( 𝐹 Fn 𝑌 → ( 𝑧 ∈ ran 𝐹 ↔ ∃ 𝑦 ∈ 𝑌 ( 𝐹 ‘ 𝑦 ) = 𝑧 ) ) |
| 82 |
10 81
|
syl |
⊢ ( 𝐹 : 𝑌 ⟶ 𝑋 → ( 𝑧 ∈ ran 𝐹 ↔ ∃ 𝑦 ∈ 𝑌 ( 𝐹 ‘ 𝑦 ) = 𝑧 ) ) |
| 83 |
82
|
3ad2ant3 |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝑧 ∈ ran 𝐹 ↔ ∃ 𝑦 ∈ 𝑌 ( 𝐹 ‘ 𝑦 ) = 𝑧 ) ) |
| 84 |
83
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → ( 𝑧 ∈ ran 𝐹 ↔ ∃ 𝑦 ∈ 𝑌 ( 𝐹 ‘ 𝑦 ) = 𝑧 ) ) |
| 85 |
72
|
ad2antrr |
⊢ ( ( ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) ∧ 𝑦 ∈ 𝑌 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑥 ) → Fun 𝐹 ) |
| 86 |
85 73
|
jctir |
⊢ ( ( ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) ∧ 𝑦 ∈ 𝑌 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑥 ) → ( Fun 𝐹 ∧ ( ◡ 𝐹 “ 𝑥 ) ⊆ dom 𝐹 ) ) |
| 87 |
57
|
ad2antrr |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) → Fun 𝐹 ) |
| 88 |
87
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) ∧ 𝑦 ∈ 𝑌 ) → Fun 𝐹 ) |
| 89 |
45
|
3ad2ant3 |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → dom 𝐹 = 𝑌 ) |
| 90 |
89
|
ad3antrrr |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → dom 𝐹 = 𝑌 ) |
| 91 |
90
|
eleq2d |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → ( 𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝑌 ) ) |
| 92 |
91
|
biimpar |
⊢ ( ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) ∧ 𝑦 ∈ 𝑌 ) → 𝑦 ∈ dom 𝐹 ) |
| 93 |
|
fvimacnv |
⊢ ( ( Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑥 ↔ 𝑦 ∈ ( ◡ 𝐹 “ 𝑥 ) ) ) |
| 94 |
88 92 93
|
syl2anc |
⊢ ( ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) ∧ 𝑦 ∈ 𝑌 ) → ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑥 ↔ 𝑦 ∈ ( ◡ 𝐹 “ 𝑥 ) ) ) |
| 95 |
94
|
biimpa |
⊢ ( ( ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) ∧ 𝑦 ∈ 𝑌 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑥 ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑥 ) ) |
| 96 |
|
funfvima2 |
⊢ ( ( Fun 𝐹 ∧ ( ◡ 𝐹 “ 𝑥 ) ⊆ dom 𝐹 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ∈ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ) ) |
| 97 |
86 95 96
|
sylc |
⊢ ( ( ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) ∧ 𝑦 ∈ 𝑌 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ∈ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ) |
| 98 |
97
|
ex |
⊢ ( ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) ∧ 𝑦 ∈ 𝑌 ) → ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑥 → ( 𝐹 ‘ 𝑦 ) ∈ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ) ) |
| 99 |
|
eleq1 |
⊢ ( ( 𝐹 ‘ 𝑦 ) = 𝑧 → ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑥 ↔ 𝑧 ∈ 𝑥 ) ) |
| 100 |
|
eleq1 |
⊢ ( ( 𝐹 ‘ 𝑦 ) = 𝑧 → ( ( 𝐹 ‘ 𝑦 ) ∈ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ↔ 𝑧 ∈ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ) ) |
| 101 |
99 100
|
imbi12d |
⊢ ( ( 𝐹 ‘ 𝑦 ) = 𝑧 → ( ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑥 → ( 𝐹 ‘ 𝑦 ) ∈ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ) ↔ ( 𝑧 ∈ 𝑥 → 𝑧 ∈ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ) ) ) |
| 102 |
98 101
|
syl5ibcom |
⊢ ( ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) ∧ 𝑦 ∈ 𝑌 ) → ( ( 𝐹 ‘ 𝑦 ) = 𝑧 → ( 𝑧 ∈ 𝑥 → 𝑧 ∈ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ) ) ) |
| 103 |
102
|
rexlimdva |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → ( ∃ 𝑦 ∈ 𝑌 ( 𝐹 ‘ 𝑦 ) = 𝑧 → ( 𝑧 ∈ 𝑥 → 𝑧 ∈ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ) ) ) |
| 104 |
84 103
|
sylbid |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → ( 𝑧 ∈ ran 𝐹 → ( 𝑧 ∈ 𝑥 → 𝑧 ∈ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ) ) ) |
| 105 |
104
|
impcomd |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → ( ( 𝑧 ∈ 𝑥 ∧ 𝑧 ∈ ran 𝐹 ) → 𝑧 ∈ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ) ) |
| 106 |
80 105
|
syl5bi |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → ( 𝑧 ∈ ( 𝑥 ∩ ran 𝐹 ) → 𝑧 ∈ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ) ) |
| 107 |
106
|
ssrdv |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → ( 𝑥 ∩ ran 𝐹 ) ⊆ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ) |
| 108 |
79 107
|
eqssd |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) = ( 𝑥 ∩ ran 𝐹 ) ) |
| 109 |
|
filin |
⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝑥 ∈ 𝐿 ∧ ran 𝐹 ∈ 𝐿 ) → ( 𝑥 ∩ ran 𝐹 ) ∈ 𝐿 ) |
| 110 |
109
|
3exp |
⊢ ( 𝐿 ∈ ( Fil ‘ 𝑋 ) → ( 𝑥 ∈ 𝐿 → ( ran 𝐹 ∈ 𝐿 → ( 𝑥 ∩ ran 𝐹 ) ∈ 𝐿 ) ) ) |
| 111 |
110
|
com23 |
⊢ ( 𝐿 ∈ ( Fil ‘ 𝑋 ) → ( ran 𝐹 ∈ 𝐿 → ( 𝑥 ∈ 𝐿 → ( 𝑥 ∩ ran 𝐹 ) ∈ 𝐿 ) ) ) |
| 112 |
111
|
3ad2ant2 |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( ran 𝐹 ∈ 𝐿 → ( 𝑥 ∈ 𝐿 → ( 𝑥 ∩ ran 𝐹 ) ∈ 𝐿 ) ) ) |
| 113 |
112
|
imp31 |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) → ( 𝑥 ∩ ran 𝐹 ) ∈ 𝐿 ) |
| 114 |
113
|
adantr |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → ( 𝑥 ∩ ran 𝐹 ) ∈ 𝐿 ) |
| 115 |
108 114
|
eqeltrd |
⊢ ( ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) ∧ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ∧ 𝑡 ⊆ 𝑋 ) ) → ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ∈ 𝐿 ) |
| 116 |
115
|
exp32 |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) → ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 → ( 𝑡 ⊆ 𝑋 → ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ∈ 𝐿 ) ) ) |
| 117 |
|
imaeq2 |
⊢ ( 𝑠 = ( ◡ 𝐹 “ 𝑥 ) → ( 𝐹 “ 𝑠 ) = ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ) |
| 118 |
117
|
sseq1d |
⊢ ( 𝑠 = ( ◡ 𝐹 “ 𝑥 ) → ( ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ↔ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 ) ) |
| 119 |
117
|
eleq1d |
⊢ ( 𝑠 = ( ◡ 𝐹 “ 𝑥 ) → ( ( 𝐹 “ 𝑠 ) ∈ 𝐿 ↔ ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ∈ 𝐿 ) ) |
| 120 |
119
|
imbi2d |
⊢ ( 𝑠 = ( ◡ 𝐹 “ 𝑥 ) → ( ( 𝑡 ⊆ 𝑋 → ( 𝐹 “ 𝑠 ) ∈ 𝐿 ) ↔ ( 𝑡 ⊆ 𝑋 → ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ∈ 𝐿 ) ) ) |
| 121 |
118 120
|
imbi12d |
⊢ ( 𝑠 = ( ◡ 𝐹 “ 𝑥 ) → ( ( ( 𝐹 “ 𝑠 ) ⊆ 𝑡 → ( 𝑡 ⊆ 𝑋 → ( 𝐹 “ 𝑠 ) ∈ 𝐿 ) ) ↔ ( ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ⊆ 𝑡 → ( 𝑡 ⊆ 𝑋 → ( 𝐹 “ ( ◡ 𝐹 “ 𝑥 ) ) ∈ 𝐿 ) ) ) ) |
| 122 |
116 121
|
syl5ibrcom |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ 𝑥 ∈ 𝐿 ) → ( 𝑠 = ( ◡ 𝐹 “ 𝑥 ) → ( ( 𝐹 “ 𝑠 ) ⊆ 𝑡 → ( 𝑡 ⊆ 𝑋 → ( 𝐹 “ 𝑠 ) ∈ 𝐿 ) ) ) ) |
| 123 |
122
|
rexlimdva |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( ∃ 𝑥 ∈ 𝐿 𝑠 = ( ◡ 𝐹 “ 𝑥 ) → ( ( 𝐹 “ 𝑠 ) ⊆ 𝑡 → ( 𝑡 ⊆ 𝑋 → ( 𝐹 “ 𝑠 ) ∈ 𝐿 ) ) ) ) |
| 124 |
70 123
|
syl5bi |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) → ( ( 𝐹 “ 𝑠 ) ⊆ 𝑡 → ( 𝑡 ⊆ 𝑋 → ( 𝐹 “ 𝑠 ) ∈ 𝐿 ) ) ) ) |
| 125 |
124
|
imp44 |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ ( ( 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ∧ ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) ∧ 𝑡 ⊆ 𝑋 ) ) → ( 𝐹 “ 𝑠 ) ∈ 𝐿 ) |
| 126 |
|
simprr |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ ( ( 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ∧ ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) ∧ 𝑡 ⊆ 𝑋 ) ) → 𝑡 ⊆ 𝑋 ) |
| 127 |
|
simprlr |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ ( ( 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ∧ ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) ∧ 𝑡 ⊆ 𝑋 ) ) → ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) |
| 128 |
|
filss |
⊢ ( ( 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ ( ( 𝐹 “ 𝑠 ) ∈ 𝐿 ∧ 𝑡 ⊆ 𝑋 ∧ ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) ) → 𝑡 ∈ 𝐿 ) |
| 129 |
68 125 126 127 128
|
syl13anc |
⊢ ( ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) ∧ ( ( 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ∧ ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) ∧ 𝑡 ⊆ 𝑋 ) ) → 𝑡 ∈ 𝐿 ) |
| 130 |
129
|
exp44 |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) → ( ( 𝐹 “ 𝑠 ) ⊆ 𝑡 → ( 𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿 ) ) ) ) |
| 131 |
130
|
rexlimdv |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( ∃ 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ( 𝐹 “ 𝑠 ) ⊆ 𝑡 → ( 𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿 ) ) ) |
| 132 |
131
|
impcomd |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) → 𝑡 ∈ 𝐿 ) ) |
| 133 |
67 132
|
impbid |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( 𝑡 ∈ 𝐿 ↔ ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) ) ) |
| 134 |
2
|
adantr |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → 𝑋 ∈ 𝐿 ) |
| 135 |
|
rnelfmlem |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ∈ ( fBas ‘ 𝑌 ) ) |
| 136 |
|
simpl3 |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → 𝐹 : 𝑌 ⟶ 𝑋 ) |
| 137 |
|
elfm |
⊢ ( ( 𝑋 ∈ 𝐿 ∧ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ∈ ( fBas ‘ 𝑌 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝑡 ∈ ( ( 𝑋 FilMap 𝐹 ) ‘ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ) ↔ ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) ) ) |
| 138 |
134 135 136 137
|
syl3anc |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( 𝑡 ∈ ( ( 𝑋 FilMap 𝐹 ) ‘ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ) ↔ ( 𝑡 ⊆ 𝑋 ∧ ∃ 𝑠 ∈ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ( 𝐹 “ 𝑠 ) ⊆ 𝑡 ) ) ) |
| 139 |
133 138
|
bitr4d |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( 𝑡 ∈ 𝐿 ↔ 𝑡 ∈ ( ( 𝑋 FilMap 𝐹 ) ‘ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ) ) ) |
| 140 |
139
|
eqrdv |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → 𝐿 = ( ( 𝑋 FilMap 𝐹 ) ‘ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ) ) |
| 141 |
7
|
adantr |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( 𝑋 FilMap 𝐹 ) Fn ( fBas ‘ 𝑌 ) ) |
| 142 |
|
fnfvelrn |
⊢ ( ( ( 𝑋 FilMap 𝐹 ) Fn ( fBas ‘ 𝑌 ) ∧ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ∈ ( fBas ‘ 𝑌 ) ) → ( ( 𝑋 FilMap 𝐹 ) ‘ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ) ∈ ran ( 𝑋 FilMap 𝐹 ) ) |
| 143 |
141 135 142
|
syl2anc |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → ( ( 𝑋 FilMap 𝐹 ) ‘ ran ( 𝑥 ∈ 𝐿 ↦ ( ◡ 𝐹 “ 𝑥 ) ) ) ∈ ran ( 𝑋 FilMap 𝐹 ) ) |
| 144 |
140 143
|
eqeltrd |
⊢ ( ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) ∧ ran 𝐹 ∈ 𝐿 ) → 𝐿 ∈ ran ( 𝑋 FilMap 𝐹 ) ) |
| 145 |
144
|
ex |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( ran 𝐹 ∈ 𝐿 → 𝐿 ∈ ran ( 𝑋 FilMap 𝐹 ) ) ) |
| 146 |
33 145
|
impbid |
⊢ ( ( 𝑌 ∈ 𝐴 ∧ 𝐿 ∈ ( Fil ‘ 𝑋 ) ∧ 𝐹 : 𝑌 ⟶ 𝑋 ) → ( 𝐿 ∈ ran ( 𝑋 FilMap 𝐹 ) ↔ ran 𝐹 ∈ 𝐿 ) ) |