Step |
Hyp |
Ref |
Expression |
1 |
|
ffsrn.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝑊 ) |
2 |
|
ffsrn.0 |
⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) |
3 |
|
ffsrn.1 |
⊢ ( 𝜑 → Fun 𝐹 ) |
4 |
|
ffsrn.2 |
⊢ ( 𝜑 → ( 𝐹 supp 𝑍 ) ∈ Fin ) |
5 |
|
dfdm4 |
⊢ dom 𝐹 = ran ◡ 𝐹 |
6 |
|
dfrn4 |
⊢ ran ◡ 𝐹 = ( ◡ 𝐹 “ V ) |
7 |
5 6
|
eqtri |
⊢ dom 𝐹 = ( ◡ 𝐹 “ V ) |
8 |
|
df-fn |
⊢ ( 𝐹 Fn ( ◡ 𝐹 “ V ) ↔ ( Fun 𝐹 ∧ dom 𝐹 = ( ◡ 𝐹 “ V ) ) ) |
9 |
|
fnresdm |
⊢ ( 𝐹 Fn ( ◡ 𝐹 “ V ) → ( 𝐹 ↾ ( ◡ 𝐹 “ V ) ) = 𝐹 ) |
10 |
8 9
|
sylbir |
⊢ ( ( Fun 𝐹 ∧ dom 𝐹 = ( ◡ 𝐹 “ V ) ) → ( 𝐹 ↾ ( ◡ 𝐹 “ V ) ) = 𝐹 ) |
11 |
3 7 10
|
sylancl |
⊢ ( 𝜑 → ( 𝐹 ↾ ( ◡ 𝐹 “ V ) ) = 𝐹 ) |
12 |
|
imaundi |
⊢ ( ◡ 𝐹 “ ( ( V ∖ { 𝑍 } ) ∪ { 𝑍 } ) ) = ( ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ∪ ( ◡ 𝐹 “ { 𝑍 } ) ) |
13 |
12
|
reseq2i |
⊢ ( 𝐹 ↾ ( ◡ 𝐹 “ ( ( V ∖ { 𝑍 } ) ∪ { 𝑍 } ) ) ) = ( 𝐹 ↾ ( ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ∪ ( ◡ 𝐹 “ { 𝑍 } ) ) ) |
14 |
|
undif1 |
⊢ ( ( V ∖ { 𝑍 } ) ∪ { 𝑍 } ) = ( V ∪ { 𝑍 } ) |
15 |
|
ssv |
⊢ { 𝑍 } ⊆ V |
16 |
|
ssequn2 |
⊢ ( { 𝑍 } ⊆ V ↔ ( V ∪ { 𝑍 } ) = V ) |
17 |
15 16
|
mpbi |
⊢ ( V ∪ { 𝑍 } ) = V |
18 |
14 17
|
eqtri |
⊢ ( ( V ∖ { 𝑍 } ) ∪ { 𝑍 } ) = V |
19 |
18
|
imaeq2i |
⊢ ( ◡ 𝐹 “ ( ( V ∖ { 𝑍 } ) ∪ { 𝑍 } ) ) = ( ◡ 𝐹 “ V ) |
20 |
19
|
reseq2i |
⊢ ( 𝐹 ↾ ( ◡ 𝐹 “ ( ( V ∖ { 𝑍 } ) ∪ { 𝑍 } ) ) ) = ( 𝐹 ↾ ( ◡ 𝐹 “ V ) ) |
21 |
|
resundi |
⊢ ( 𝐹 ↾ ( ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ∪ ( ◡ 𝐹 “ { 𝑍 } ) ) ) = ( ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ∪ ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ) |
22 |
13 20 21
|
3eqtr3i |
⊢ ( 𝐹 ↾ ( ◡ 𝐹 “ V ) ) = ( ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ∪ ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ) |
23 |
11 22
|
eqtr3di |
⊢ ( 𝜑 → 𝐹 = ( ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ∪ ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ) ) |
24 |
23
|
rneqd |
⊢ ( 𝜑 → ran 𝐹 = ran ( ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ∪ ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ) ) |
25 |
|
rnun |
⊢ ran ( ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ∪ ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ) = ( ran ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ∪ ran ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ) |
26 |
24 25
|
eqtrdi |
⊢ ( 𝜑 → ran 𝐹 = ( ran ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ∪ ran ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ) ) |
27 |
|
suppimacnv |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ) → ( 𝐹 supp 𝑍 ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
28 |
2 1 27
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 supp 𝑍 ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
29 |
28 4
|
eqeltrrd |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ∈ Fin ) |
30 |
|
cnvexg |
⊢ ( 𝐹 ∈ 𝑉 → ◡ 𝐹 ∈ V ) |
31 |
|
imaexg |
⊢ ( ◡ 𝐹 ∈ V → ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ∈ V ) |
32 |
2 30 31
|
3syl |
⊢ ( 𝜑 → ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ∈ V ) |
33 |
|
cnvimass |
⊢ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ dom 𝐹 |
34 |
|
fores |
⊢ ( ( Fun 𝐹 ∧ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ dom 𝐹 ) → ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) : ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) –onto→ ( 𝐹 “ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ) |
35 |
3 33 34
|
sylancl |
⊢ ( 𝜑 → ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) : ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) –onto→ ( 𝐹 “ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ) |
36 |
|
fofn |
⊢ ( ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) : ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) –onto→ ( 𝐹 “ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) → ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) Fn ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
37 |
35 36
|
syl |
⊢ ( 𝜑 → ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) Fn ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
38 |
|
fnrndomg |
⊢ ( ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ∈ V → ( ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) Fn ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) → ran ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ≼ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ) |
39 |
32 37 38
|
sylc |
⊢ ( 𝜑 → ran ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ≼ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
40 |
|
domfi |
⊢ ( ( ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ∈ Fin ∧ ran ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ≼ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) → ran ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ∈ Fin ) |
41 |
29 39 40
|
syl2anc |
⊢ ( 𝜑 → ran ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ∈ Fin ) |
42 |
|
snfi |
⊢ { 𝑍 } ∈ Fin |
43 |
|
df-ima |
⊢ ( 𝐹 “ ( ◡ 𝐹 “ { 𝑍 } ) ) = ran ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) |
44 |
|
funimacnv |
⊢ ( Fun 𝐹 → ( 𝐹 “ ( ◡ 𝐹 “ { 𝑍 } ) ) = ( { 𝑍 } ∩ ran 𝐹 ) ) |
45 |
3 44
|
syl |
⊢ ( 𝜑 → ( 𝐹 “ ( ◡ 𝐹 “ { 𝑍 } ) ) = ( { 𝑍 } ∩ ran 𝐹 ) ) |
46 |
43 45
|
eqtr3id |
⊢ ( 𝜑 → ran ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) = ( { 𝑍 } ∩ ran 𝐹 ) ) |
47 |
|
inss1 |
⊢ ( { 𝑍 } ∩ ran 𝐹 ) ⊆ { 𝑍 } |
48 |
46 47
|
eqsstrdi |
⊢ ( 𝜑 → ran ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ⊆ { 𝑍 } ) |
49 |
|
ssfi |
⊢ ( ( { 𝑍 } ∈ Fin ∧ ran ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ⊆ { 𝑍 } ) → ran ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ∈ Fin ) |
50 |
42 48 49
|
sylancr |
⊢ ( 𝜑 → ran ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ∈ Fin ) |
51 |
|
unfi |
⊢ ( ( ran ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ∈ Fin ∧ ran ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ∈ Fin ) → ( ran ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ∪ ran ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ) ∈ Fin ) |
52 |
41 50 51
|
syl2anc |
⊢ ( 𝜑 → ( ran ( 𝐹 ↾ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ∪ ran ( 𝐹 ↾ ( ◡ 𝐹 “ { 𝑍 } ) ) ) ∈ Fin ) |
53 |
26 52
|
eqeltrd |
⊢ ( 𝜑 → ran 𝐹 ∈ Fin ) |