Metamath Proof Explorer

Theorem fpwwe2lem5

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 fpwwe2lem5 
|- ( ( ph /\ C R ( M ` B ) ) -> ( C e. X /\ C e. Y /\ ( `' M ` C ) = ( `' N ` C ) ) )

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 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 relopabiv 
 |-  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 ) ) )