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Theorem fpwwe2lem7 8258
Description: Lemma for fpwwe2 8265. (Contributed by Mario Carneiro, 18-May-2015.)
Hypotheses
Ref Expression
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 )
fpwwe2lem9.x  |-  ( ph  ->  X W R )
fpwwe2lem9.y  |-  ( ph  ->  Y W S )
fpwwe2lem9.m  |-  M  = OrdIso
( R ,  X
)
fpwwe2lem9.n  |-  N  = OrdIso
( S ,  Y
)
fpwwe2lem7.1  |-  ( ph  ->  B  e.  dom  M
)
fpwwe2lem7.2  |-  ( ph  ->  B  e.  dom  N
)
fpwwe2lem7.3  |-  ( ph  ->  ( M  |`  B )  =  ( N  |`  B ) )
Assertion
Ref Expression
fpwwe2lem7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( C S ( N `  B )  /\  ( D R ( M `  B )  ->  ( C R D  <->  C S D ) ) ) )
Distinct variable groups:    y, u, B    u, r, x, y, F    X, r, u, x, y    M, r, u, x, y    N, r, u, x, y    ph, r, u, x, y    A, r, x    R, r, u, x, y    Y, r, u, x, y    S, r, u, x, y    W, r, u, x, y
Allowed substitution hints:    A( y, u)    B( x, r)    C( x, y, u, r)    D( x, y, u, r)

Proof of Theorem fpwwe2lem7
StepHypRef Expression
1 fpwwe2lem9.y . . . . . . . 8  |-  ( ph  ->  Y W S )
2 fpwwe2.1 . . . . . . . . . 10  |-  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 ) ) }
32relopabi 4811 . . . . . . . . 9  |-  Rel  W
43brrelexi 4729 . . . . . . . 8  |-  ( Y W S  ->  Y  e.  _V )
51, 4syl 15 . . . . . . 7  |-  ( ph  ->  Y  e.  _V )
6 fpwwe2.2 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  _V )
72, 6fpwwe2lem2 8254 . . . . . . . . . 10  |-  ( 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 ) ) ) )
81, 7mpbid 201 . . . . . . . . 9  |-  ( 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 ) ) )
98simprd 449 . . . . . . . 8  |-  ( ph  ->  ( S  We  Y  /\  A. y  e.  Y  [. ( `' S " { y } )  /  u ]. (
u F ( S  i^i  ( u  X.  u ) ) )  =  y ) )
109simpld 445 . . . . . . 7  |-  ( ph  ->  S  We  Y )
11 fpwwe2lem9.n . . . . . . . 8  |-  N  = OrdIso
( S ,  Y
)
1211oiiso 7252 . . . . . . 7  |-  ( ( Y  e.  _V  /\  S  We  Y )  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
135, 10, 12syl2anc 642 . . . . . 6  |-  ( ph  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
1413adantr 451 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
15 isof1o 5822 . . . . 5  |-  ( N 
Isom  _E  ,  S  ( dom  N ,  Y
)  ->  N : dom  N -1-1-onto-> Y )
1614, 15syl 15 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  N : dom  N -1-1-onto-> Y )
17 fpwwe2.3 . . . . . 6  |-  ( (
ph  /\  ( x  C_  A  /\  r  C_  ( x  X.  x
)  /\  r  We  x ) )  -> 
( x F r )  e.  A )
18 fpwwe2lem9.x . . . . . 6  |-  ( ph  ->  X W R )
19 fpwwe2lem9.m . . . . . 6  |-  M  = OrdIso
( R ,  X
)
20 fpwwe2lem7.1 . . . . . 6  |-  ( ph  ->  B  e.  dom  M
)
21 fpwwe2lem7.2 . . . . . 6  |-  ( ph  ->  B  e.  dom  N
)
22 fpwwe2lem7.3 . . . . . 6  |-  ( ph  ->  ( M  |`  B )  =  ( N  |`  B ) )
232, 6, 17, 18, 1, 19, 11, 20, 21, 22fpwwe2lem6 8257 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( C  e.  X  /\  C  e.  Y  /\  ( `' M `  C )  =  ( `' N `  C ) ) )
2423simp2d 968 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  Y )
25 f1ocnvfv2 5793 . . . 4  |-  ( ( N : dom  N -1-1-onto-> Y  /\  C  e.  Y
)  ->  ( N `  ( `' N `  C ) )  =  C )
2616, 24, 25syl2anc 642 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( N `  ( `' N `  C )
)  =  C )
2723simp3d 969 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  =  ( `' N `  C ) )
283brrelexi 4729 . . . . . . . . . . . 12  |-  ( X W R  ->  X  e.  _V )
2918, 28syl 15 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  _V )
302, 6fpwwe2lem2 8254 . . . . . . . . . . . . . 14  |-  ( 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 ) ) ) )
3118, 30mpbid 201 . . . . . . . . . . . . 13  |-  ( 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 ) ) )
3231simprd 449 . . . . . . . . . . . 12  |-  ( ph  ->  ( R  We  X  /\  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
3332simpld 445 . . . . . . . . . . 11  |-  ( ph  ->  R  We  X )
3419oiiso 7252 . . . . . . . . . . 11  |-  ( ( X  e.  _V  /\  R  We  X )  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
3529, 33, 34syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
3635adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
37 isof1o 5822 . . . . . . . . 9  |-  ( M 
Isom  _E  ,  R  ( dom  M ,  X
)  ->  M : dom  M -1-1-onto-> X )
3836, 37syl 15 . . . . . . . 8  |-  ( (
ph  /\  C R
( M `  B
) )  ->  M : dom  M -1-1-onto-> X )
3923simp1d 967 . . . . . . . 8  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  X )
40 f1ocnvfv2 5793 . . . . . . . 8  |-  ( ( M : dom  M -1-1-onto-> X  /\  C  e.  X
)  ->  ( M `  ( `' M `  C ) )  =  C )
4138, 39, 40syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M `  ( `' M `  C )
)  =  C )
42 simpr 447 . . . . . . 7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C R ( M `  B ) )
4341, 42eqbrtrd 4043 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M `  ( `' M `  C )
) R ( M `
 B ) )
44 f1ocnv 5485 . . . . . . . . 9  |-  ( M : dom  M -1-1-onto-> X  ->  `' M : X -1-1-onto-> dom  M
)
45 f1of 5472 . . . . . . . . 9  |-  ( `' M : X -1-1-onto-> dom  M  ->  `' M : X --> dom  M
)
4638, 44, 453syl 18 . . . . . . . 8  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' M : X --> dom  M
)
47 ffvelrn 5663 . . . . . . . 8  |-  ( ( `' M : X --> dom  M  /\  C  e.  X
)  ->  ( `' M `  C )  e.  dom  M )
4846, 39, 47syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  e.  dom  M )
4920adantr 451 . . . . . . 7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  B  e.  dom  M )
50 isorel 5823 . . . . . . 7  |-  ( ( 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
) ) )
5136, 48, 49, 50syl12anc 1180 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' M `  C )  _E  B  <->  ( M `  ( `' M `  C ) ) R ( M `
 B ) ) )
5243, 51mpbird 223 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  _E  B )
5327, 52eqbrtrrd 4045 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' N `  C )  _E  B )
54 f1ocnv 5485 . . . . . . 7  |-  ( N : dom  N -1-1-onto-> Y  ->  `' N : Y -1-1-onto-> dom  N
)
55 f1of 5472 . . . . . . 7  |-  ( `' N : Y -1-1-onto-> dom  N  ->  `' N : Y --> dom  N
)
5616, 54, 553syl 18 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' N : Y --> dom  N
)
57 ffvelrn 5663 . . . . . 6  |-  ( ( `' N : Y --> dom  N  /\  C  e.  Y
)  ->  ( `' N `  C )  e.  dom  N )
5856, 24, 57syl2anc 642 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' N `  C )  e.  dom  N )
5921adantr 451 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  B  e.  dom  N )
60 isorel 5823 . . . . 5  |-  ( ( 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
) ) )
6114, 58, 59, 60syl12anc 1180 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' N `  C )  _E  B  <->  ( N `  ( `' N `  C ) ) S ( N `
 B ) ) )
6253, 61mpbid 201 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( N `  ( `' N `  C )
) S ( N `
 B ) )
6326, 62eqbrtrrd 4045 . 2  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C S ( N `  B ) )
6427adantrr 697 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  ( `' M `  C )  =  ( `' N `  C ) )
652, 6, 17, 18, 1, 19, 11, 20, 21, 22fpwwe2lem6 8257 . . . . . . 7  |-  ( (
ph  /\  D R
( M `  B
) )  ->  ( D  e.  X  /\  D  e.  Y  /\  ( `' M `  D )  =  ( `' N `  D ) ) )
6665simp3d 969 . . . . . 6  |-  ( (
ph  /\  D R
( M `  B
) )  ->  ( `' M `  D )  =  ( `' N `  D ) )
6766adantrl 696 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  ( `' M `  D )  =  ( `' N `  D ) )
6864, 67breq12d 4036 . . . 4  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  ( ( `' M `  C )  _E  ( `' M `  D )  <->  ( `' N `  C )  _E  ( `' N `  D ) ) )
6935adantr 451 . . . . . 6  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  M  Isom  _E  ,  R  ( dom  M ,  X ) )
70 isocnv 5827 . . . . . 6  |-  ( M 
Isom  _E  ,  R  ( dom  M ,  X
)  ->  `' M  Isom  R ,  _E  ( X ,  dom  M ) )
7169, 70syl 15 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  `' M  Isom  R ,  _E  ( X ,  dom  M ) )
7239adantrr 697 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  C  e.  X
)
7331simpld 445 . . . . . . . . . 10  |-  ( ph  ->  ( X  C_  A  /\  R  C_  ( X  X.  X ) ) )
7473simprd 449 . . . . . . . . 9  |-  ( ph  ->  R  C_  ( X  X.  X ) )
7574ssbrd 4064 . . . . . . . 8  |-  ( ph  ->  ( D R ( M `  B )  ->  D ( X  X.  X ) ( M `  B ) ) )
7675imp 418 . . . . . . 7  |-  ( (
ph  /\  D R
( M `  B
) )  ->  D
( X  X.  X
) ( M `  B ) )
77 brxp 4720 . . . . . . . 8  |-  ( D ( X  X.  X
) ( M `  B )  <->  ( D  e.  X  /\  ( M `  B )  e.  X ) )
7877simplbi 446 . . . . . . 7  |-  ( D ( X  X.  X
) ( M `  B )  ->  D  e.  X )
7976, 78syl 15 . . . . . 6  |-  ( (
ph  /\  D R
( M `  B
) )  ->  D  e.  X )
8079adantrl 696 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  D  e.  X
)
81 isorel 5823 . . . . 5  |-  ( ( `' M  Isom  R ,  _E  ( X ,  dom  M )  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( C R D  <-> 
( `' M `  C )  _E  ( `' M `  D ) ) )
8271, 72, 80, 81syl12anc 1180 . . . 4  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  ( C R D  <->  ( `' M `  C )  _E  ( `' M `  D ) ) )
8313adantr 451 . . . . . 6  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  N  Isom  _E  ,  S  ( dom  N ,  Y ) )
84 isocnv 5827 . . . . . 6  |-  ( N 
Isom  _E  ,  S  ( dom  N ,  Y
)  ->  `' N  Isom  S ,  _E  ( Y ,  dom  N ) )
8583, 84syl 15 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  `' N  Isom  S ,  _E  ( Y ,  dom  N ) )
8624adantrr 697 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  C  e.  Y
)
8765simp2d 968 . . . . . 6  |-  ( (
ph  /\  D R
( M `  B
) )  ->  D  e.  Y )
8887adantrl 696 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  D  e.  Y
)
89 isorel 5823 . . . . 5  |-  ( ( `' N  Isom  S ,  _E  ( Y ,  dom  N )  /\  ( C  e.  Y  /\  D  e.  Y ) )  -> 
( C S D  <-> 
( `' N `  C )  _E  ( `' N `  D ) ) )
9085, 86, 88, 89syl12anc 1180 . . . 4  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  ( C S D  <->  ( `' N `  C )  _E  ( `' N `  D ) ) )
9168, 82, 903bitr4d 276 . . 3  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  ( C R D  <->  C S D ) )
9291expr 598 . 2  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( D R ( M `  B )  ->  ( C R D  <->  C S D ) ) )
9363, 92jca 518 1  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( C S ( N `  B )  /\  ( D R ( M `  B )  ->  ( C R D  <->  C S D ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788   [.wsbc 2991    i^i cin 3151    C_ wss 3152   {csn 3640   class class class wbr 4023   {copab 4076    _E cep 4303    We wwe 4351    X. cxp 4687   `'ccnv 4688   dom cdm 4689    |` cres 4691   "cima 4692   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256  (class class class)co 5858  OrdIsocoi 7224
This theorem is referenced by:  fpwwe2lem8  8259
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-riota 6304  df-recs 6388  df-oi 7225
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