Step |
Hyp |
Ref |
Expression |
1 |
|
lrrec.1 |
|- R = { <. x , y >. | x e. ( ( _L ` y ) u. ( _R ` y ) ) } |
2 |
|
df-fr |
|- ( R Fr No <-> A. a ( ( a C_ No /\ a =/= (/) ) -> E. p e. a A. q e. a -. q R p ) ) |
3 |
|
bdayfun |
|- Fun bday |
4 |
|
imassrn |
|- ( bday " a ) C_ ran bday |
5 |
|
bdayrn |
|- ran bday = On |
6 |
4 5
|
sseqtri |
|- ( bday " a ) C_ On |
7 |
|
fvex |
|- ( bday ` q ) e. _V |
8 |
7
|
jctr |
|- ( q e. a -> ( q e. a /\ ( bday ` q ) e. _V ) ) |
9 |
8
|
eximi |
|- ( E. q q e. a -> E. q ( q e. a /\ ( bday ` q ) e. _V ) ) |
10 |
|
n0 |
|- ( a =/= (/) <-> E. q q e. a ) |
11 |
|
df-rex |
|- ( E. q e. a ( bday ` q ) e. _V <-> E. q ( q e. a /\ ( bday ` q ) e. _V ) ) |
12 |
9 10 11
|
3imtr4i |
|- ( a =/= (/) -> E. q e. a ( bday ` q ) e. _V ) |
13 |
|
isset |
|- ( ( bday ` q ) e. _V <-> E. p p = ( bday ` q ) ) |
14 |
|
eqcom |
|- ( p = ( bday ` q ) <-> ( bday ` q ) = p ) |
15 |
14
|
exbii |
|- ( E. p p = ( bday ` q ) <-> E. p ( bday ` q ) = p ) |
16 |
13 15
|
bitri |
|- ( ( bday ` q ) e. _V <-> E. p ( bday ` q ) = p ) |
17 |
16
|
rexbii |
|- ( E. q e. a ( bday ` q ) e. _V <-> E. q e. a E. p ( bday ` q ) = p ) |
18 |
|
rexcom4 |
|- ( E. q e. a E. p ( bday ` q ) = p <-> E. p E. q e. a ( bday ` q ) = p ) |
19 |
17 18
|
bitri |
|- ( E. q e. a ( bday ` q ) e. _V <-> E. p E. q e. a ( bday ` q ) = p ) |
20 |
12 19
|
sylib |
|- ( a =/= (/) -> E. p E. q e. a ( bday ` q ) = p ) |
21 |
20
|
adantl |
|- ( ( a C_ No /\ a =/= (/) ) -> E. p E. q e. a ( bday ` q ) = p ) |
22 |
|
bdayfn |
|- bday Fn No |
23 |
|
fvelimab |
|- ( ( bday Fn No /\ a C_ No ) -> ( p e. ( bday " a ) <-> E. q e. a ( bday ` q ) = p ) ) |
24 |
22 23
|
mpan |
|- ( a C_ No -> ( p e. ( bday " a ) <-> E. q e. a ( bday ` q ) = p ) ) |
25 |
24
|
adantr |
|- ( ( a C_ No /\ a =/= (/) ) -> ( p e. ( bday " a ) <-> E. q e. a ( bday ` q ) = p ) ) |
26 |
25
|
exbidv |
|- ( ( a C_ No /\ a =/= (/) ) -> ( E. p p e. ( bday " a ) <-> E. p E. q e. a ( bday ` q ) = p ) ) |
27 |
21 26
|
mpbird |
|- ( ( a C_ No /\ a =/= (/) ) -> E. p p e. ( bday " a ) ) |
28 |
|
n0 |
|- ( ( bday " a ) =/= (/) <-> E. p p e. ( bday " a ) ) |
29 |
27 28
|
sylibr |
|- ( ( a C_ No /\ a =/= (/) ) -> ( bday " a ) =/= (/) ) |
30 |
|
onint |
|- ( ( ( bday " a ) C_ On /\ ( bday " a ) =/= (/) ) -> |^| ( bday " a ) e. ( bday " a ) ) |
31 |
6 29 30
|
sylancr |
|- ( ( a C_ No /\ a =/= (/) ) -> |^| ( bday " a ) e. ( bday " a ) ) |
32 |
|
fvelima |
|- ( ( Fun bday /\ |^| ( bday " a ) e. ( bday " a ) ) -> E. p e. a ( bday ` p ) = |^| ( bday " a ) ) |
33 |
3 31 32
|
sylancr |
|- ( ( a C_ No /\ a =/= (/) ) -> E. p e. a ( bday ` p ) = |^| ( bday " a ) ) |
34 |
|
fnfvima |
|- ( ( bday Fn No /\ a C_ No /\ q e. a ) -> ( bday ` q ) e. ( bday " a ) ) |
35 |
22 34
|
mp3an1 |
|- ( ( a C_ No /\ q e. a ) -> ( bday ` q ) e. ( bday " a ) ) |
36 |
35
|
adantlr |
|- ( ( ( a C_ No /\ a =/= (/) ) /\ q e. a ) -> ( bday ` q ) e. ( bday " a ) ) |
37 |
|
onnmin |
|- ( ( ( bday " a ) C_ On /\ ( bday ` q ) e. ( bday " a ) ) -> -. ( bday ` q ) e. |^| ( bday " a ) ) |
38 |
6 36 37
|
sylancr |
|- ( ( ( a C_ No /\ a =/= (/) ) /\ q e. a ) -> -. ( bday ` q ) e. |^| ( bday " a ) ) |
39 |
38
|
ralrimiva |
|- ( ( a C_ No /\ a =/= (/) ) -> A. q e. a -. ( bday ` q ) e. |^| ( bday " a ) ) |
40 |
|
eleq2 |
|- ( ( bday ` p ) = |^| ( bday " a ) -> ( ( bday ` q ) e. ( bday ` p ) <-> ( bday ` q ) e. |^| ( bday " a ) ) ) |
41 |
40
|
notbid |
|- ( ( bday ` p ) = |^| ( bday " a ) -> ( -. ( bday ` q ) e. ( bday ` p ) <-> -. ( bday ` q ) e. |^| ( bday " a ) ) ) |
42 |
41
|
ralbidv |
|- ( ( bday ` p ) = |^| ( bday " a ) -> ( A. q e. a -. ( bday ` q ) e. ( bday ` p ) <-> A. q e. a -. ( bday ` q ) e. |^| ( bday " a ) ) ) |
43 |
39 42
|
syl5ibrcom |
|- ( ( a C_ No /\ a =/= (/) ) -> ( ( bday ` p ) = |^| ( bday " a ) -> A. q e. a -. ( bday ` q ) e. ( bday ` p ) ) ) |
44 |
43
|
reximdv |
|- ( ( a C_ No /\ a =/= (/) ) -> ( E. p e. a ( bday ` p ) = |^| ( bday " a ) -> E. p e. a A. q e. a -. ( bday ` q ) e. ( bday ` p ) ) ) |
45 |
33 44
|
mpd |
|- ( ( a C_ No /\ a =/= (/) ) -> E. p e. a A. q e. a -. ( bday ` q ) e. ( bday ` p ) ) |
46 |
|
simpll |
|- ( ( ( a C_ No /\ a =/= (/) ) /\ ( p e. a /\ q e. a ) ) -> a C_ No ) |
47 |
|
simprr |
|- ( ( ( a C_ No /\ a =/= (/) ) /\ ( p e. a /\ q e. a ) ) -> q e. a ) |
48 |
46 47
|
sseldd |
|- ( ( ( a C_ No /\ a =/= (/) ) /\ ( p e. a /\ q e. a ) ) -> q e. No ) |
49 |
|
simprl |
|- ( ( ( a C_ No /\ a =/= (/) ) /\ ( p e. a /\ q e. a ) ) -> p e. a ) |
50 |
46 49
|
sseldd |
|- ( ( ( a C_ No /\ a =/= (/) ) /\ ( p e. a /\ q e. a ) ) -> p e. No ) |
51 |
1
|
lrrecval2 |
|- ( ( q e. No /\ p e. No ) -> ( q R p <-> ( bday ` q ) e. ( bday ` p ) ) ) |
52 |
48 50 51
|
syl2anc |
|- ( ( ( a C_ No /\ a =/= (/) ) /\ ( p e. a /\ q e. a ) ) -> ( q R p <-> ( bday ` q ) e. ( bday ` p ) ) ) |
53 |
52
|
notbid |
|- ( ( ( a C_ No /\ a =/= (/) ) /\ ( p e. a /\ q e. a ) ) -> ( -. q R p <-> -. ( bday ` q ) e. ( bday ` p ) ) ) |
54 |
53
|
anassrs |
|- ( ( ( ( a C_ No /\ a =/= (/) ) /\ p e. a ) /\ q e. a ) -> ( -. q R p <-> -. ( bday ` q ) e. ( bday ` p ) ) ) |
55 |
54
|
ralbidva |
|- ( ( ( a C_ No /\ a =/= (/) ) /\ p e. a ) -> ( A. q e. a -. q R p <-> A. q e. a -. ( bday ` q ) e. ( bday ` p ) ) ) |
56 |
55
|
rexbidva |
|- ( ( a C_ No /\ a =/= (/) ) -> ( E. p e. a A. q e. a -. q R p <-> E. p e. a A. q e. a -. ( bday ` q ) e. ( bday ` p ) ) ) |
57 |
45 56
|
mpbird |
|- ( ( a C_ No /\ a =/= (/) ) -> E. p e. a A. q e. a -. q R p ) |
58 |
2 57
|
mpgbir |
|- R Fr No |