Step |
Hyp |
Ref |
Expression |
1 |
|
infpssrlem.a |
|- ( ph -> B C_ A ) |
2 |
|
infpssrlem.c |
|- ( ph -> F : B -1-1-onto-> A ) |
3 |
|
infpssrlem.d |
|- ( ph -> C e. ( A \ B ) ) |
4 |
|
infpssrlem.e |
|- G = ( rec ( `' F , C ) |` _om ) |
5 |
|
frfnom |
|- ( rec ( `' F , C ) |` _om ) Fn _om |
6 |
4
|
fneq1i |
|- ( G Fn _om <-> ( rec ( `' F , C ) |` _om ) Fn _om ) |
7 |
5 6
|
mpbir |
|- G Fn _om |
8 |
7
|
a1i |
|- ( ph -> G Fn _om ) |
9 |
|
fveq2 |
|- ( c = (/) -> ( G ` c ) = ( G ` (/) ) ) |
10 |
9
|
eleq1d |
|- ( c = (/) -> ( ( G ` c ) e. A <-> ( G ` (/) ) e. A ) ) |
11 |
|
fveq2 |
|- ( c = b -> ( G ` c ) = ( G ` b ) ) |
12 |
11
|
eleq1d |
|- ( c = b -> ( ( G ` c ) e. A <-> ( G ` b ) e. A ) ) |
13 |
|
fveq2 |
|- ( c = suc b -> ( G ` c ) = ( G ` suc b ) ) |
14 |
13
|
eleq1d |
|- ( c = suc b -> ( ( G ` c ) e. A <-> ( G ` suc b ) e. A ) ) |
15 |
1 2 3 4
|
infpssrlem1 |
|- ( ph -> ( G ` (/) ) = C ) |
16 |
3
|
eldifad |
|- ( ph -> C e. A ) |
17 |
15 16
|
eqeltrd |
|- ( ph -> ( G ` (/) ) e. A ) |
18 |
1
|
adantr |
|- ( ( ph /\ ( G ` b ) e. A ) -> B C_ A ) |
19 |
|
f1ocnv |
|- ( F : B -1-1-onto-> A -> `' F : A -1-1-onto-> B ) |
20 |
|
f1of |
|- ( `' F : A -1-1-onto-> B -> `' F : A --> B ) |
21 |
2 19 20
|
3syl |
|- ( ph -> `' F : A --> B ) |
22 |
21
|
ffvelrnda |
|- ( ( ph /\ ( G ` b ) e. A ) -> ( `' F ` ( G ` b ) ) e. B ) |
23 |
18 22
|
sseldd |
|- ( ( ph /\ ( G ` b ) e. A ) -> ( `' F ` ( G ` b ) ) e. A ) |
24 |
1 2 3 4
|
infpssrlem2 |
|- ( b e. _om -> ( G ` suc b ) = ( `' F ` ( G ` b ) ) ) |
25 |
24
|
eleq1d |
|- ( b e. _om -> ( ( G ` suc b ) e. A <-> ( `' F ` ( G ` b ) ) e. A ) ) |
26 |
23 25
|
syl5ibr |
|- ( b e. _om -> ( ( ph /\ ( G ` b ) e. A ) -> ( G ` suc b ) e. A ) ) |
27 |
26
|
expd |
|- ( b e. _om -> ( ph -> ( ( G ` b ) e. A -> ( G ` suc b ) e. A ) ) ) |
28 |
10 12 14 17 27
|
finds2 |
|- ( c e. _om -> ( ph -> ( G ` c ) e. A ) ) |
29 |
28
|
com12 |
|- ( ph -> ( c e. _om -> ( G ` c ) e. A ) ) |
30 |
29
|
ralrimiv |
|- ( ph -> A. c e. _om ( G ` c ) e. A ) |
31 |
|
ffnfv |
|- ( G : _om --> A <-> ( G Fn _om /\ A. c e. _om ( G ` c ) e. A ) ) |
32 |
8 30 31
|
sylanbrc |
|- ( ph -> G : _om --> A ) |