Step |
Hyp |
Ref |
Expression |
1 |
|
mbfres2.1 |
|- ( ph -> F : A --> RR ) |
2 |
|
mbfres2.2 |
|- ( ph -> ( F |` B ) e. MblFn ) |
3 |
|
mbfres2.3 |
|- ( ph -> ( F |` C ) e. MblFn ) |
4 |
|
mbfres2.4 |
|- ( ph -> ( B u. C ) = A ) |
5 |
4
|
reseq2d |
|- ( ph -> ( F |` ( B u. C ) ) = ( F |` A ) ) |
6 |
|
ffn |
|- ( F : A --> RR -> F Fn A ) |
7 |
|
fnresdm |
|- ( F Fn A -> ( F |` A ) = F ) |
8 |
1 6 7
|
3syl |
|- ( ph -> ( F |` A ) = F ) |
9 |
5 8
|
eqtr2d |
|- ( ph -> F = ( F |` ( B u. C ) ) ) |
10 |
9
|
adantr |
|- ( ( ph /\ x e. ran (,) ) -> F = ( F |` ( B u. C ) ) ) |
11 |
|
resundi |
|- ( F |` ( B u. C ) ) = ( ( F |` B ) u. ( F |` C ) ) |
12 |
10 11
|
eqtrdi |
|- ( ( ph /\ x e. ran (,) ) -> F = ( ( F |` B ) u. ( F |` C ) ) ) |
13 |
12
|
cnveqd |
|- ( ( ph /\ x e. ran (,) ) -> `' F = `' ( ( F |` B ) u. ( F |` C ) ) ) |
14 |
|
cnvun |
|- `' ( ( F |` B ) u. ( F |` C ) ) = ( `' ( F |` B ) u. `' ( F |` C ) ) |
15 |
13 14
|
eqtrdi |
|- ( ( ph /\ x e. ran (,) ) -> `' F = ( `' ( F |` B ) u. `' ( F |` C ) ) ) |
16 |
15
|
imaeq1d |
|- ( ( ph /\ x e. ran (,) ) -> ( `' F " x ) = ( ( `' ( F |` B ) u. `' ( F |` C ) ) " x ) ) |
17 |
|
imaundir |
|- ( ( `' ( F |` B ) u. `' ( F |` C ) ) " x ) = ( ( `' ( F |` B ) " x ) u. ( `' ( F |` C ) " x ) ) |
18 |
16 17
|
eqtrdi |
|- ( ( ph /\ x e. ran (,) ) -> ( `' F " x ) = ( ( `' ( F |` B ) " x ) u. ( `' ( F |` C ) " x ) ) ) |
19 |
|
ssun1 |
|- B C_ ( B u. C ) |
20 |
19 4
|
sseqtrid |
|- ( ph -> B C_ A ) |
21 |
1 20
|
fssresd |
|- ( ph -> ( F |` B ) : B --> RR ) |
22 |
|
ismbf |
|- ( ( F |` B ) : B --> RR -> ( ( F |` B ) e. MblFn <-> A. x e. ran (,) ( `' ( F |` B ) " x ) e. dom vol ) ) |
23 |
21 22
|
syl |
|- ( ph -> ( ( F |` B ) e. MblFn <-> A. x e. ran (,) ( `' ( F |` B ) " x ) e. dom vol ) ) |
24 |
2 23
|
mpbid |
|- ( ph -> A. x e. ran (,) ( `' ( F |` B ) " x ) e. dom vol ) |
25 |
24
|
r19.21bi |
|- ( ( ph /\ x e. ran (,) ) -> ( `' ( F |` B ) " x ) e. dom vol ) |
26 |
|
ssun2 |
|- C C_ ( B u. C ) |
27 |
26 4
|
sseqtrid |
|- ( ph -> C C_ A ) |
28 |
1 27
|
fssresd |
|- ( ph -> ( F |` C ) : C --> RR ) |
29 |
|
ismbf |
|- ( ( F |` C ) : C --> RR -> ( ( F |` C ) e. MblFn <-> A. x e. ran (,) ( `' ( F |` C ) " x ) e. dom vol ) ) |
30 |
28 29
|
syl |
|- ( ph -> ( ( F |` C ) e. MblFn <-> A. x e. ran (,) ( `' ( F |` C ) " x ) e. dom vol ) ) |
31 |
3 30
|
mpbid |
|- ( ph -> A. x e. ran (,) ( `' ( F |` C ) " x ) e. dom vol ) |
32 |
31
|
r19.21bi |
|- ( ( ph /\ x e. ran (,) ) -> ( `' ( F |` C ) " x ) e. dom vol ) |
33 |
|
unmbl |
|- ( ( ( `' ( F |` B ) " x ) e. dom vol /\ ( `' ( F |` C ) " x ) e. dom vol ) -> ( ( `' ( F |` B ) " x ) u. ( `' ( F |` C ) " x ) ) e. dom vol ) |
34 |
25 32 33
|
syl2anc |
|- ( ( ph /\ x e. ran (,) ) -> ( ( `' ( F |` B ) " x ) u. ( `' ( F |` C ) " x ) ) e. dom vol ) |
35 |
18 34
|
eqeltrd |
|- ( ( ph /\ x e. ran (,) ) -> ( `' F " x ) e. dom vol ) |
36 |
35
|
ralrimiva |
|- ( ph -> A. x e. ran (,) ( `' F " x ) e. dom vol ) |
37 |
|
ismbf |
|- ( F : A --> RR -> ( F e. MblFn <-> A. x e. ran (,) ( `' F " x ) e. dom vol ) ) |
38 |
1 37
|
syl |
|- ( ph -> ( F e. MblFn <-> A. x e. ran (,) ( `' F " x ) e. dom vol ) ) |
39 |
36 38
|
mpbird |
|- ( ph -> F e. MblFn ) |