Step |
Hyp |
Ref |
Expression |
1 |
|
sdomdom |
|- ( A ~< B -> A ~<_ B ) |
2 |
|
brdomi |
|- ( A ~<_ B -> E. f f : A -1-1-> B ) |
3 |
1 2
|
syl |
|- ( A ~< B -> E. f f : A -1-1-> B ) |
4 |
|
vex |
|- f e. _V |
5 |
4
|
rnex |
|- ran f e. _V |
6 |
|
f1f1orn |
|- ( f : A -1-1-> B -> f : A -1-1-onto-> ran f ) |
7 |
6
|
adantl |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> f : A -1-1-onto-> ran f ) |
8 |
|
f1of1 |
|- ( f : A -1-1-onto-> ran f -> f : A -1-1-> ran f ) |
9 |
7 8
|
syl |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> f : A -1-1-> ran f ) |
10 |
|
f1dom3g |
|- ( ( f e. _V /\ ran f e. _V /\ f : A -1-1-> ran f ) -> A ~<_ ran f ) |
11 |
4 5 9 10
|
mp3an12i |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> A ~<_ ran f ) |
12 |
|
sdomnen |
|- ( A ~< B -> -. A ~~ B ) |
13 |
12
|
adantr |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> -. A ~~ B ) |
14 |
|
ssdif0 |
|- ( B C_ ran f <-> ( B \ ran f ) = (/) ) |
15 |
|
simplr |
|- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> f : A -1-1-> B ) |
16 |
|
f1f |
|- ( f : A -1-1-> B -> f : A --> B ) |
17 |
16
|
frnd |
|- ( f : A -1-1-> B -> ran f C_ B ) |
18 |
15 17
|
syl |
|- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> ran f C_ B ) |
19 |
|
simpr |
|- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> B C_ ran f ) |
20 |
18 19
|
eqssd |
|- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> ran f = B ) |
21 |
|
dff1o5 |
|- ( f : A -1-1-onto-> B <-> ( f : A -1-1-> B /\ ran f = B ) ) |
22 |
15 20 21
|
sylanbrc |
|- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> f : A -1-1-onto-> B ) |
23 |
|
f1oen3g |
|- ( ( f e. _V /\ f : A -1-1-onto-> B ) -> A ~~ B ) |
24 |
4 22 23
|
sylancr |
|- ( ( ( A ~< B /\ f : A -1-1-> B ) /\ B C_ ran f ) -> A ~~ B ) |
25 |
24
|
ex |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> ( B C_ ran f -> A ~~ B ) ) |
26 |
14 25
|
biimtrrid |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> ( ( B \ ran f ) = (/) -> A ~~ B ) ) |
27 |
13 26
|
mtod |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> -. ( B \ ran f ) = (/) ) |
28 |
|
neq0 |
|- ( -. ( B \ ran f ) = (/) <-> E. w w e. ( B \ ran f ) ) |
29 |
27 28
|
sylib |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> E. w w e. ( B \ ran f ) ) |
30 |
|
snssi |
|- ( w e. ( B \ ran f ) -> { w } C_ ( B \ ran f ) ) |
31 |
|
relsdom |
|- Rel ~< |
32 |
31
|
brrelex1i |
|- ( A ~< B -> A e. _V ) |
33 |
32
|
adantr |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> A e. _V ) |
34 |
|
vex |
|- w e. _V |
35 |
|
en2sn |
|- ( ( A e. _V /\ w e. _V ) -> { A } ~~ { w } ) |
36 |
33 34 35
|
sylancl |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> { A } ~~ { w } ) |
37 |
31
|
brrelex2i |
|- ( A ~< B -> B e. _V ) |
38 |
37
|
adantr |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> B e. _V ) |
39 |
|
difexg |
|- ( B e. _V -> ( B \ ran f ) e. _V ) |
40 |
|
snfi |
|- { w } e. Fin |
41 |
|
ssdomfi2 |
|- ( ( { w } e. Fin /\ ( B \ ran f ) e. _V /\ { w } C_ ( B \ ran f ) ) -> { w } ~<_ ( B \ ran f ) ) |
42 |
40 41
|
mp3an1 |
|- ( ( ( B \ ran f ) e. _V /\ { w } C_ ( B \ ran f ) ) -> { w } ~<_ ( B \ ran f ) ) |
43 |
42
|
ex |
|- ( ( B \ ran f ) e. _V -> ( { w } C_ ( B \ ran f ) -> { w } ~<_ ( B \ ran f ) ) ) |
44 |
38 39 43
|
3syl |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> ( { w } C_ ( B \ ran f ) -> { w } ~<_ ( B \ ran f ) ) ) |
45 |
|
endom |
|- ( { A } ~~ { w } -> { A } ~<_ { w } ) |
46 |
|
domtrfi |
|- ( ( { w } e. Fin /\ { A } ~<_ { w } /\ { w } ~<_ ( B \ ran f ) ) -> { A } ~<_ ( B \ ran f ) ) |
47 |
40 46
|
mp3an1 |
|- ( ( { A } ~<_ { w } /\ { w } ~<_ ( B \ ran f ) ) -> { A } ~<_ ( B \ ran f ) ) |
48 |
45 47
|
sylan |
|- ( ( { A } ~~ { w } /\ { w } ~<_ ( B \ ran f ) ) -> { A } ~<_ ( B \ ran f ) ) |
49 |
36 44 48
|
syl6an |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> ( { w } C_ ( B \ ran f ) -> { A } ~<_ ( B \ ran f ) ) ) |
50 |
30 49
|
syl5 |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> ( w e. ( B \ ran f ) -> { A } ~<_ ( B \ ran f ) ) ) |
51 |
50
|
exlimdv |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> ( E. w w e. ( B \ ran f ) -> { A } ~<_ ( B \ ran f ) ) ) |
52 |
29 51
|
mpd |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> { A } ~<_ ( B \ ran f ) ) |
53 |
|
disjdif |
|- ( ran f i^i ( B \ ran f ) ) = (/) |
54 |
53
|
a1i |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> ( ran f i^i ( B \ ran f ) ) = (/) ) |
55 |
|
undom |
|- ( ( ( A ~<_ ran f /\ { A } ~<_ ( B \ ran f ) ) /\ ( ran f i^i ( B \ ran f ) ) = (/) ) -> ( A u. { A } ) ~<_ ( ran f u. ( B \ ran f ) ) ) |
56 |
11 52 54 55
|
syl21anc |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> ( A u. { A } ) ~<_ ( ran f u. ( B \ ran f ) ) ) |
57 |
|
df-suc |
|- suc A = ( A u. { A } ) |
58 |
57
|
a1i |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> suc A = ( A u. { A } ) ) |
59 |
|
undif2 |
|- ( ran f u. ( B \ ran f ) ) = ( ran f u. B ) |
60 |
17
|
adantl |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> ran f C_ B ) |
61 |
|
ssequn1 |
|- ( ran f C_ B <-> ( ran f u. B ) = B ) |
62 |
60 61
|
sylib |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> ( ran f u. B ) = B ) |
63 |
59 62
|
eqtr2id |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> B = ( ran f u. ( B \ ran f ) ) ) |
64 |
56 58 63
|
3brtr4d |
|- ( ( A ~< B /\ f : A -1-1-> B ) -> suc A ~<_ B ) |
65 |
3 64
|
exlimddv |
|- ( A ~< B -> suc A ~<_ B ) |