Step |
Hyp |
Ref |
Expression |
1 |
|
ctex |
|- ( A ~<_ _om -> A e. _V ) |
2 |
1
|
adantl |
|- ( ( F Fn A /\ A ~<_ _om ) -> A e. _V ) |
3 |
|
fndm |
|- ( F Fn A -> dom F = A ) |
4 |
3
|
eleq1d |
|- ( F Fn A -> ( dom F e. _V <-> A e. _V ) ) |
5 |
4
|
adantr |
|- ( ( F Fn A /\ A ~<_ _om ) -> ( dom F e. _V <-> A e. _V ) ) |
6 |
2 5
|
mpbird |
|- ( ( F Fn A /\ A ~<_ _om ) -> dom F e. _V ) |
7 |
|
fnfun |
|- ( F Fn A -> Fun F ) |
8 |
7
|
adantr |
|- ( ( F Fn A /\ A ~<_ _om ) -> Fun F ) |
9 |
|
funrnex |
|- ( dom F e. _V -> ( Fun F -> ran F e. _V ) ) |
10 |
6 8 9
|
sylc |
|- ( ( F Fn A /\ A ~<_ _om ) -> ran F e. _V ) |
11 |
2 10
|
xpexd |
|- ( ( F Fn A /\ A ~<_ _om ) -> ( A X. ran F ) e. _V ) |
12 |
|
simpl |
|- ( ( F Fn A /\ A ~<_ _om ) -> F Fn A ) |
13 |
|
dffn3 |
|- ( F Fn A <-> F : A --> ran F ) |
14 |
12 13
|
sylib |
|- ( ( F Fn A /\ A ~<_ _om ) -> F : A --> ran F ) |
15 |
|
fssxp |
|- ( F : A --> ran F -> F C_ ( A X. ran F ) ) |
16 |
14 15
|
syl |
|- ( ( F Fn A /\ A ~<_ _om ) -> F C_ ( A X. ran F ) ) |
17 |
|
ssdomg |
|- ( ( A X. ran F ) e. _V -> ( F C_ ( A X. ran F ) -> F ~<_ ( A X. ran F ) ) ) |
18 |
11 16 17
|
sylc |
|- ( ( F Fn A /\ A ~<_ _om ) -> F ~<_ ( A X. ran F ) ) |
19 |
|
xpdom1g |
|- ( ( ran F e. _V /\ A ~<_ _om ) -> ( A X. ran F ) ~<_ ( _om X. ran F ) ) |
20 |
10 19
|
sylancom |
|- ( ( F Fn A /\ A ~<_ _om ) -> ( A X. ran F ) ~<_ ( _om X. ran F ) ) |
21 |
|
omex |
|- _om e. _V |
22 |
|
fnrndomg |
|- ( A e. _V -> ( F Fn A -> ran F ~<_ A ) ) |
23 |
2 12 22
|
sylc |
|- ( ( F Fn A /\ A ~<_ _om ) -> ran F ~<_ A ) |
24 |
|
domtr |
|- ( ( ran F ~<_ A /\ A ~<_ _om ) -> ran F ~<_ _om ) |
25 |
23 24
|
sylancom |
|- ( ( F Fn A /\ A ~<_ _om ) -> ran F ~<_ _om ) |
26 |
|
xpdom2g |
|- ( ( _om e. _V /\ ran F ~<_ _om ) -> ( _om X. ran F ) ~<_ ( _om X. _om ) ) |
27 |
21 25 26
|
sylancr |
|- ( ( F Fn A /\ A ~<_ _om ) -> ( _om X. ran F ) ~<_ ( _om X. _om ) ) |
28 |
|
domtr |
|- ( ( ( A X. ran F ) ~<_ ( _om X. ran F ) /\ ( _om X. ran F ) ~<_ ( _om X. _om ) ) -> ( A X. ran F ) ~<_ ( _om X. _om ) ) |
29 |
20 27 28
|
syl2anc |
|- ( ( F Fn A /\ A ~<_ _om ) -> ( A X. ran F ) ~<_ ( _om X. _om ) ) |
30 |
|
xpomen |
|- ( _om X. _om ) ~~ _om |
31 |
|
domentr |
|- ( ( ( A X. ran F ) ~<_ ( _om X. _om ) /\ ( _om X. _om ) ~~ _om ) -> ( A X. ran F ) ~<_ _om ) |
32 |
29 30 31
|
sylancl |
|- ( ( F Fn A /\ A ~<_ _om ) -> ( A X. ran F ) ~<_ _om ) |
33 |
|
domtr |
|- ( ( F ~<_ ( A X. ran F ) /\ ( A X. ran F ) ~<_ _om ) -> F ~<_ _om ) |
34 |
18 32 33
|
syl2anc |
|- ( ( F Fn A /\ A ~<_ _om ) -> F ~<_ _om ) |