Metamath Proof Explorer

Theorem fpwwe2lem6

Description: Lemma for fpwwe2 . (Contributed by Mario Carneiro, 18-May-2015) (Revised by AV, 20-Jul-2024)

Ref Expression
Hypotheses 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 ) ) }
fpwwe2.2 
|- ( ph -> A e. V )
fpwwe2.3 
|- ( ( ph /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x F r ) e. A )
fpwwe2lem8.x 
|- ( ph -> X W R )
fpwwe2lem8.y 
|- ( ph -> Y W S )
fpwwe2lem8.m 
|- M = OrdIso ( R , X )
fpwwe2lem8.n 
|- N = OrdIso ( S , Y )
fpwwe2lem5.1 
|- ( ph -> B e. dom M )
fpwwe2lem5.2 
|- ( ph -> B e. dom N )
fpwwe2lem5.3 
|- ( ph -> ( M |` B ) = ( N |` B ) )
Assertion fpwwe2lem6 
|- ( ( ph /\ C R ( M ` B ) ) -> ( C S ( N ` B ) /\ ( D R ( M ` B ) -> ( C R D <-> C S D ) ) ) )

Proof

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 relopabiv 
 |-  Rel W
12 11 brrelex1i 
 |-  ( Y W S -> Y e. _V )
13 5 12 syl 
 |-  ( ph -> Y e. _V )
14 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 ) ) ) )
15 5 14 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 ) ) )
16 15 simprld 
 |-  ( ph -> S We Y )
17 7 oiiso 
 |-  ( ( Y e. _V /\ S We Y ) -> N Isom _E , S ( dom N , Y ) )
18 13 16 17 syl2anc 
 |-  ( ph -> N Isom _E , S ( dom N , Y ) )
19 18 adantr 
 |-  ( ( ph /\ C R ( M ` B ) ) -> N Isom _E , S ( dom N , Y ) )
20 isof1o 
 |-  ( N Isom _E , S ( dom N , Y ) -> N : dom N -1-1-onto-> Y )
21 19 20 syl 
 |-  ( ( ph /\ C R ( M ` B ) ) -> N : dom N -1-1-onto-> Y )
22 1 2 3 4 5 6 7 8 9 10 fpwwe2lem5 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( C e. X /\ C e. Y /\ ( `' M ` C ) = ( `' N ` C ) ) )
23 22 simp2d 
 |-  ( ( ph /\ C R ( M ` B ) ) -> C e. Y )
24 f1ocnvfv2 
 |-  ( ( N : dom N -1-1-onto-> Y /\ C e. Y ) -> ( N ` ( `' N ` C ) ) = C )
25 21 23 24 syl2anc 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( N ` ( `' N ` C ) ) = C )
26 22 simp3d 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) = ( `' N ` C ) )
27 11 brrelex1i 
 |-  ( X W R -> X e. _V )
28 4 27 syl 
 |-  ( ph -> X e. _V )
29 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 ) ) ) )
30 4 29 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 ) ) )
31 30 simprld 
 |-  ( ph -> R We X )
32 6 oiiso 
 |-  ( ( X e. _V /\ R We X ) -> M Isom _E , R ( dom M , X ) )
33 28 31 32 syl2anc 
 |-  ( ph -> M Isom _E , R ( dom M , X ) )
34 33 adantr 
 |-  ( ( ph /\ C R ( M ` B ) ) -> M Isom _E , R ( dom M , X ) )
35 isof1o 
 |-  ( M Isom _E , R ( dom M , X ) -> M : dom M -1-1-onto-> X )
36 34 35 syl 
 |-  ( ( ph /\ C R ( M ` B ) ) -> M : dom M -1-1-onto-> X )
37 22 simp1d 
 |-  ( ( ph /\ C R ( M ` B ) ) -> C e. X )
38 f1ocnvfv2 
 |-  ( ( M : dom M -1-1-onto-> X /\ C e. X ) -> ( M ` ( `' M ` C ) ) = C )
39 36 37 38 syl2anc 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( M ` ( `' M ` C ) ) = C )
40 simpr 
 |-  ( ( ph /\ C R ( M ` B ) ) -> C R ( M ` B ) )
41 39 40 eqbrtrd 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( M ` ( `' M ` C ) ) R ( M ` B ) )
42 f1ocnv 
 |-  ( M : dom M -1-1-onto-> X -> `' M : X -1-1-onto-> dom M )
43 f1of 
 |-  ( `' M : X -1-1-onto-> dom M -> `' M : X --> dom M )
44 36 42 43 3syl 
 |-  ( ( ph /\ C R ( M ` B ) ) -> `' M : X --> dom M )
45 44 37 ffvelrnd 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) e. dom M )
46 8 adantr 
 |-  ( ( ph /\ C R ( M ` B ) ) -> B e. dom M )
47 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 ) ) )
48 34 45 46 47 syl12anc 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( ( `' M ` C ) _E B <-> ( M ` ( `' M ` C ) ) R ( M ` B ) ) )
49 41 48 mpbird 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) _E B )
50 26 49 eqbrtrrd 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( `' N ` C ) _E B )
51 f1ocnv 
 |-  ( N : dom N -1-1-onto-> Y -> `' N : Y -1-1-onto-> dom N )
52 f1of 
 |-  ( `' N : Y -1-1-onto-> dom N -> `' N : Y --> dom N )
53 21 51 52 3syl 
 |-  ( ( ph /\ C R ( M ` B ) ) -> `' N : Y --> dom N )
54 53 23 ffvelrnd 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( `' N ` C ) e. dom N )
55 9 adantr 
 |-  ( ( ph /\ C R ( M ` B ) ) -> B e. dom N )
56 isorel 
 |-  ( ( N Isom _E , S ( dom N , Y ) /\ ( ( `' N ` C ) e. dom N /\ B e. dom N ) ) -> ( ( `' N ` C ) _E B <-> ( N ` ( `' N ` C ) ) S ( N ` B ) ) )
57 19 54 55 56 syl12anc 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( ( `' N ` C ) _E B <-> ( N ` ( `' N ` C ) ) S ( N ` B ) ) )
58 50 57 mpbid 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( N ` ( `' N ` C ) ) S ( N ` B ) )
59 25 58 eqbrtrrd 
 |-  ( ( ph /\ C R ( M ` B ) ) -> C S ( N ` B ) )
60 26 adantrr 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> ( `' M ` C ) = ( `' N ` C ) )
61 1 2 3 4 5 6 7 8 9 10 fpwwe2lem5 
 |-  ( ( ph /\ D R ( M ` B ) ) -> ( D e. X /\ D e. Y /\ ( `' M ` D ) = ( `' N ` D ) ) )
62 61 simp3d 
 |-  ( ( ph /\ D R ( M ` B ) ) -> ( `' M ` D ) = ( `' N ` D ) )
63 62 adantrl 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> ( `' M ` D ) = ( `' N ` D ) )
64 60 63 breq12d 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> ( ( `' M ` C ) _E ( `' M ` D ) <-> ( `' N ` C ) _E ( `' N ` D ) ) )
65 33 adantr 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> M Isom _E , R ( dom M , X ) )
66 isocnv 
 |-  ( M Isom _E , R ( dom M , X ) -> `' M Isom R , _E ( X , dom M ) )
67 65 66 syl 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> `' M Isom R , _E ( X , dom M ) )
68 37 adantrr 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> C e. X )
69 30 simplrd 
 |-  ( ph -> R C_ ( X X. X ) )
70 69 ssbrd 
 |-  ( ph -> ( D R ( M ` B ) -> D ( X X. X ) ( M ` B ) ) )
71 70 imp 
 |-  ( ( ph /\ D R ( M ` B ) ) -> D ( X X. X ) ( M ` B ) )
72 brxp 
 |-  ( D ( X X. X ) ( M ` B ) <-> ( D e. X /\ ( M ` B ) e. X ) )
73 72 simplbi 
 |-  ( D ( X X. X ) ( M ` B ) -> D e. X )
74 71 73 syl 
 |-  ( ( ph /\ D R ( M ` B ) ) -> D e. X )
75 74 adantrl 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> D e. X )
76 isorel 
 |-  ( ( `' M Isom R , _E ( X , dom M ) /\ ( C e. X /\ D e. X ) ) -> ( C R D <-> ( `' M ` C ) _E ( `' M ` D ) ) )
77 67 68 75 76 syl12anc 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> ( C R D <-> ( `' M ` C ) _E ( `' M ` D ) ) )
78 18 adantr 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> N Isom _E , S ( dom N , Y ) )
79 isocnv 
 |-  ( N Isom _E , S ( dom N , Y ) -> `' N Isom S , _E ( Y , dom N ) )
80 78 79 syl 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> `' N Isom S , _E ( Y , dom N ) )
81 23 adantrr 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> C e. Y )
82 61 simp2d 
 |-  ( ( ph /\ D R ( M ` B ) ) -> D e. Y )
83 82 adantrl 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> D e. Y )
84 isorel 
 |-  ( ( `' N Isom S , _E ( Y , dom N ) /\ ( C e. Y /\ D e. Y ) ) -> ( C S D <-> ( `' N ` C ) _E ( `' N ` D ) ) )
85 80 81 83 84 syl12anc 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> ( C S D <-> ( `' N ` C ) _E ( `' N ` D ) ) )
86 64 77 85 3bitr4d 
 |-  ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> ( C R D <-> C S D ) )
87 86 expr 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( D R ( M ` B ) -> ( C R D <-> C S D ) ) )
88 59 87 jca 
 |-  ( ( ph /\ C R ( M ` B ) ) -> ( C S ( N ` B ) /\ ( D R ( M ` B ) -> ( C R D <-> C S D ) ) ) )