Step |
Hyp |
Ref |
Expression |
1 |
|
fpwwe2.1 |
|- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) } |
2 |
|
fpwwe2.2 |
|- ( ph -> A e. V ) |
3 |
|
fpwwe2.3 |
|- ( ( ph /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x F r ) e. A ) |
4 |
|
fpwwe2lem8.x |
|- ( ph -> X W R ) |
5 |
|
fpwwe2lem8.y |
|- ( ph -> Y W S ) |
6 |
|
fpwwe2lem8.m |
|- M = OrdIso ( R , X ) |
7 |
|
fpwwe2lem8.n |
|- N = OrdIso ( S , Y ) |
8 |
|
fpwwe2lem5.1 |
|- ( ph -> B e. dom M ) |
9 |
|
fpwwe2lem5.2 |
|- ( ph -> B e. dom N ) |
10 |
|
fpwwe2lem5.3 |
|- ( ph -> ( M |` B ) = ( N |` B ) ) |
11 |
1 2
|
fpwwe2lem2 |
|- ( ph -> ( X W R <-> ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) ) |
12 |
4 11
|
mpbid |
|- ( ph -> ( ( X C_ A /\ R C_ ( X X. X ) ) /\ ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) ) ) |
13 |
12
|
simplrd |
|- ( ph -> R C_ ( X X. X ) ) |
14 |
13
|
ssbrd |
|- ( ph -> ( C R ( M ` B ) -> C ( X X. X ) ( M ` B ) ) ) |
15 |
|
brxp |
|- ( C ( X X. X ) ( M ` B ) <-> ( C e. X /\ ( M ` B ) e. X ) ) |
16 |
15
|
simplbi |
|- ( C ( X X. X ) ( M ` B ) -> C e. X ) |
17 |
14 16
|
syl6 |
|- ( ph -> ( C R ( M ` B ) -> C e. X ) ) |
18 |
17
|
imp |
|- ( ( ph /\ C R ( M ` B ) ) -> C e. X ) |
19 |
|
imassrn |
|- ( N " B ) C_ ran N |
20 |
1
|
relopabi |
|- Rel W |
21 |
20
|
brrelex1i |
|- ( Y W S -> Y e. _V ) |
22 |
5 21
|
syl |
|- ( ph -> Y e. _V ) |
23 |
1 2
|
fpwwe2lem2 |
|- ( ph -> ( Y W S <-> ( ( Y C_ A /\ S C_ ( Y X. Y ) ) /\ ( S We Y /\ A. y e. Y [. ( `' S " { y } ) / u ]. ( u F ( S i^i ( u X. u ) ) ) = y ) ) ) ) |
24 |
5 23
|
mpbid |
|- ( ph -> ( ( Y C_ A /\ S C_ ( Y X. Y ) ) /\ ( S We Y /\ A. y e. Y [. ( `' S " { y } ) / u ]. ( u F ( S i^i ( u X. u ) ) ) = y ) ) ) |
25 |
24
|
simprld |
|- ( ph -> S We Y ) |
26 |
7
|
oiiso |
|- ( ( Y e. _V /\ S We Y ) -> N Isom _E , S ( dom N , Y ) ) |
27 |
22 25 26
|
syl2anc |
|- ( ph -> N Isom _E , S ( dom N , Y ) ) |
28 |
27
|
adantr |
|- ( ( ph /\ C R ( M ` B ) ) -> N Isom _E , S ( dom N , Y ) ) |
29 |
|
isof1o |
|- ( N Isom _E , S ( dom N , Y ) -> N : dom N -1-1-onto-> Y ) |
30 |
28 29
|
syl |
|- ( ( ph /\ C R ( M ` B ) ) -> N : dom N -1-1-onto-> Y ) |
31 |
|
f1ofo |
|- ( N : dom N -1-1-onto-> Y -> N : dom N -onto-> Y ) |
32 |
|
forn |
|- ( N : dom N -onto-> Y -> ran N = Y ) |
33 |
30 31 32
|
3syl |
|- ( ( ph /\ C R ( M ` B ) ) -> ran N = Y ) |
34 |
19 33
|
sseqtrid |
|- ( ( ph /\ C R ( M ` B ) ) -> ( N " B ) C_ Y ) |
35 |
20
|
brrelex1i |
|- ( X W R -> X e. _V ) |
36 |
4 35
|
syl |
|- ( ph -> X e. _V ) |
37 |
12
|
simprld |
|- ( ph -> R We X ) |
38 |
6
|
oiiso |
|- ( ( X e. _V /\ R We X ) -> M Isom _E , R ( dom M , X ) ) |
39 |
36 37 38
|
syl2anc |
|- ( ph -> M Isom _E , R ( dom M , X ) ) |
40 |
39
|
adantr |
|- ( ( ph /\ C R ( M ` B ) ) -> M Isom _E , R ( dom M , X ) ) |
41 |
|
isof1o |
|- ( M Isom _E , R ( dom M , X ) -> M : dom M -1-1-onto-> X ) |
42 |
40 41
|
syl |
|- ( ( ph /\ C R ( M ` B ) ) -> M : dom M -1-1-onto-> X ) |
43 |
|
f1ocnvfv2 |
|- ( ( M : dom M -1-1-onto-> X /\ C e. X ) -> ( M ` ( `' M ` C ) ) = C ) |
44 |
42 18 43
|
syl2anc |
|- ( ( ph /\ C R ( M ` B ) ) -> ( M ` ( `' M ` C ) ) = C ) |
45 |
|
simpr |
|- ( ( ph /\ C R ( M ` B ) ) -> C R ( M ` B ) ) |
46 |
44 45
|
eqbrtrd |
|- ( ( ph /\ C R ( M ` B ) ) -> ( M ` ( `' M ` C ) ) R ( M ` B ) ) |
47 |
|
f1ocnv |
|- ( M : dom M -1-1-onto-> X -> `' M : X -1-1-onto-> dom M ) |
48 |
|
f1of |
|- ( `' M : X -1-1-onto-> dom M -> `' M : X --> dom M ) |
49 |
42 47 48
|
3syl |
|- ( ( ph /\ C R ( M ` B ) ) -> `' M : X --> dom M ) |
50 |
49 18
|
ffvelrnd |
|- ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) e. dom M ) |
51 |
8
|
adantr |
|- ( ( ph /\ C R ( M ` B ) ) -> B e. dom M ) |
52 |
|
isorel |
|- ( ( M Isom _E , R ( dom M , X ) /\ ( ( `' M ` C ) e. dom M /\ B e. dom M ) ) -> ( ( `' M ` C ) _E B <-> ( M ` ( `' M ` C ) ) R ( M ` B ) ) ) |
53 |
40 50 51 52
|
syl12anc |
|- ( ( ph /\ C R ( M ` B ) ) -> ( ( `' M ` C ) _E B <-> ( M ` ( `' M ` C ) ) R ( M ` B ) ) ) |
54 |
46 53
|
mpbird |
|- ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) _E B ) |
55 |
|
epelg |
|- ( B e. dom M -> ( ( `' M ` C ) _E B <-> ( `' M ` C ) e. B ) ) |
56 |
51 55
|
syl |
|- ( ( ph /\ C R ( M ` B ) ) -> ( ( `' M ` C ) _E B <-> ( `' M ` C ) e. B ) ) |
57 |
54 56
|
mpbid |
|- ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) e. B ) |
58 |
|
ffn |
|- ( `' M : X --> dom M -> `' M Fn X ) |
59 |
|
elpreima |
|- ( `' M Fn X -> ( C e. ( `' `' M " B ) <-> ( C e. X /\ ( `' M ` C ) e. B ) ) ) |
60 |
49 58 59
|
3syl |
|- ( ( ph /\ C R ( M ` B ) ) -> ( C e. ( `' `' M " B ) <-> ( C e. X /\ ( `' M ` C ) e. B ) ) ) |
61 |
18 57 60
|
mpbir2and |
|- ( ( ph /\ C R ( M ` B ) ) -> C e. ( `' `' M " B ) ) |
62 |
|
imacnvcnv |
|- ( `' `' M " B ) = ( M " B ) |
63 |
61 62
|
eleqtrdi |
|- ( ( ph /\ C R ( M ` B ) ) -> C e. ( M " B ) ) |
64 |
10
|
adantr |
|- ( ( ph /\ C R ( M ` B ) ) -> ( M |` B ) = ( N |` B ) ) |
65 |
64
|
rneqd |
|- ( ( ph /\ C R ( M ` B ) ) -> ran ( M |` B ) = ran ( N |` B ) ) |
66 |
|
df-ima |
|- ( M " B ) = ran ( M |` B ) |
67 |
|
df-ima |
|- ( N " B ) = ran ( N |` B ) |
68 |
65 66 67
|
3eqtr4g |
|- ( ( ph /\ C R ( M ` B ) ) -> ( M " B ) = ( N " B ) ) |
69 |
63 68
|
eleqtrd |
|- ( ( ph /\ C R ( M ` B ) ) -> C e. ( N " B ) ) |
70 |
34 69
|
sseldd |
|- ( ( ph /\ C R ( M ` B ) ) -> C e. Y ) |
71 |
64
|
cnveqd |
|- ( ( ph /\ C R ( M ` B ) ) -> `' ( M |` B ) = `' ( N |` B ) ) |
72 |
|
dff1o3 |
|- ( M : dom M -1-1-onto-> X <-> ( M : dom M -onto-> X /\ Fun `' M ) ) |
73 |
72
|
simprbi |
|- ( M : dom M -1-1-onto-> X -> Fun `' M ) |
74 |
|
funcnvres |
|- ( Fun `' M -> `' ( M |` B ) = ( `' M |` ( M " B ) ) ) |
75 |
42 73 74
|
3syl |
|- ( ( ph /\ C R ( M ` B ) ) -> `' ( M |` B ) = ( `' M |` ( M " B ) ) ) |
76 |
|
dff1o3 |
|- ( N : dom N -1-1-onto-> Y <-> ( N : dom N -onto-> Y /\ Fun `' N ) ) |
77 |
76
|
simprbi |
|- ( N : dom N -1-1-onto-> Y -> Fun `' N ) |
78 |
|
funcnvres |
|- ( Fun `' N -> `' ( N |` B ) = ( `' N |` ( N " B ) ) ) |
79 |
30 77 78
|
3syl |
|- ( ( ph /\ C R ( M ` B ) ) -> `' ( N |` B ) = ( `' N |` ( N " B ) ) ) |
80 |
71 75 79
|
3eqtr3d |
|- ( ( ph /\ C R ( M ` B ) ) -> ( `' M |` ( M " B ) ) = ( `' N |` ( N " B ) ) ) |
81 |
80
|
fveq1d |
|- ( ( ph /\ C R ( M ` B ) ) -> ( ( `' M |` ( M " B ) ) ` C ) = ( ( `' N |` ( N " B ) ) ` C ) ) |
82 |
63
|
fvresd |
|- ( ( ph /\ C R ( M ` B ) ) -> ( ( `' M |` ( M " B ) ) ` C ) = ( `' M ` C ) ) |
83 |
69
|
fvresd |
|- ( ( ph /\ C R ( M ` B ) ) -> ( ( `' N |` ( N " B ) ) ` C ) = ( `' N ` C ) ) |
84 |
81 82 83
|
3eqtr3d |
|- ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) = ( `' N ` C ) ) |
85 |
18 70 84
|
3jca |
|- ( ( ph /\ C R ( M ` B ) ) -> ( C e. X /\ C e. Y /\ ( `' M ` C ) = ( `' N ` C ) ) ) |