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Theorem fpwwe2lem7 8511
Description: Lemma for fpwwe2 8518. (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 5000 . . . . . . . . 9  |-  Rel  W
43brrelexi 4918 . . . . . . . 8  |-  ( Y W S  ->  Y  e.  _V )
51, 4syl 16 . . . . . . 7  |-  ( ph  ->  Y  e.  _V )
6 fpwwe2.2 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  _V )
72, 6fpwwe2lem2 8507 . . . . . . . . . 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 202 . . . . . . . . 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 450 . . . . . . . 8  |-  ( ph  ->  ( S  We  Y  /\  A. y  e.  Y  [. ( `' S " { y } )  /  u ]. (
u F ( S  i^i  ( u  X.  u ) ) )  =  y ) )
109simpld 446 . . . . . . 7  |-  ( ph  ->  S  We  Y )
11 fpwwe2lem9.n . . . . . . . 8  |-  N  = OrdIso
( S ,  Y
)
1211oiiso 7506 . . . . . . 7  |-  ( ( Y  e.  _V  /\  S  We  Y )  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
135, 10, 12syl2anc 643 . . . . . 6  |-  ( ph  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
1413adantr 452 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
15 isof1o 6045 . . . . 5  |-  ( N 
Isom  _E  ,  S  ( dom  N ,  Y
)  ->  N : dom  N -1-1-onto-> Y )
1614, 15syl 16 . . . 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 8510 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( C  e.  X  /\  C  e.  Y  /\  ( `' M `  C )  =  ( `' N `  C ) ) )
2423simp2d 970 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  Y )
25 f1ocnvfv2 6015 . . . 4  |-  ( ( N : dom  N -1-1-onto-> Y  /\  C  e.  Y
)  ->  ( N `  ( `' N `  C ) )  =  C )
2616, 24, 25syl2anc 643 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( N `  ( `' N `  C )
)  =  C )
2723simp3d 971 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  =  ( `' N `  C ) )
283brrelexi 4918 . . . . . . . . . . . 12  |-  ( X W R  ->  X  e.  _V )
2918, 28syl 16 . . . . . . . . . . 11  |-  ( ph  ->  X  e.  _V )
302, 6fpwwe2lem2 8507 . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . 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 450 . . . . . . . . . . . 12  |-  ( ph  ->  ( R  We  X  /\  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
3332simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  R  We  X )
3419oiiso 7506 . . . . . . . . . . 11  |-  ( ( X  e.  _V  /\  R  We  X )  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
3529, 33, 34syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
3635adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
37 isof1o 6045 . . . . . . . . 9  |-  ( M 
Isom  _E  ,  R  ( dom  M ,  X
)  ->  M : dom  M -1-1-onto-> X )
3836, 37syl 16 . . . . . . . 8  |-  ( (
ph  /\  C R
( M `  B
) )  ->  M : dom  M -1-1-onto-> X )
3923simp1d 969 . . . . . . . 8  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  X )
40 f1ocnvfv2 6015 . . . . . . . 8  |-  ( ( M : dom  M -1-1-onto-> X  /\  C  e.  X
)  ->  ( M `  ( `' M `  C ) )  =  C )
4138, 39, 40syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M `  ( `' M `  C )
)  =  C )
42 simpr 448 . . . . . . 7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C R ( M `  B ) )
4341, 42eqbrtrd 4232 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M `  ( `' M `  C )
) R ( M `
 B ) )
44 f1ocnv 5687 . . . . . . . . 9  |-  ( M : dom  M -1-1-onto-> X  ->  `' M : X -1-1-onto-> dom  M
)
45 f1of 5674 . . . . . . . . 9  |-  ( `' M : X -1-1-onto-> dom  M  ->  `' M : X --> dom  M
)
4638, 44, 453syl 19 . . . . . . . 8  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' M : X --> dom  M
)
4746, 39ffvelrnd 5871 . . . . . . 7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  e.  dom  M )
4820adantr 452 . . . . . . 7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  B  e.  dom  M )
49 isorel 6046 . . . . . . 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
) ) )
5036, 47, 48, 49syl12anc 1182 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' M `  C )  _E  B  <->  ( M `  ( `' M `  C ) ) R ( M `
 B ) ) )
5143, 50mpbird 224 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  _E  B )
5227, 51eqbrtrrd 4234 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' N `  C )  _E  B )
53 f1ocnv 5687 . . . . . . 7  |-  ( N : dom  N -1-1-onto-> Y  ->  `' N : Y -1-1-onto-> dom  N
)
54 f1of 5674 . . . . . . 7  |-  ( `' N : Y -1-1-onto-> dom  N  ->  `' N : Y --> dom  N
)
5516, 53, 543syl 19 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' N : Y --> dom  N
)
5655, 24ffvelrnd 5871 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' N `  C )  e.  dom  N )
5721adantr 452 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  B  e.  dom  N )
58 isorel 6046 . . . . 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
) ) )
5914, 56, 57, 58syl12anc 1182 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' N `  C )  _E  B  <->  ( N `  ( `' N `  C ) ) S ( N `
 B ) ) )
6052, 59mpbid 202 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( N `  ( `' N `  C )
) S ( N `
 B ) )
6126, 60eqbrtrrd 4234 . 2  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C S ( N `  B ) )
6227adantrr 698 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  ( `' M `  C )  =  ( `' N `  C ) )
632, 6, 17, 18, 1, 19, 11, 20, 21, 22fpwwe2lem6 8510 . . . . . . 7  |-  ( (
ph  /\  D R
( M `  B
) )  ->  ( D  e.  X  /\  D  e.  Y  /\  ( `' M `  D )  =  ( `' N `  D ) ) )
6463simp3d 971 . . . . . 6  |-  ( (
ph  /\  D R
( M `  B
) )  ->  ( `' M `  D )  =  ( `' N `  D ) )
6564adantrl 697 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  ( `' M `  D )  =  ( `' N `  D ) )
6662, 65breq12d 4225 . . . 4  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  ( ( `' M `  C )  _E  ( `' M `  D )  <->  ( `' N `  C )  _E  ( `' N `  D ) ) )
6735adantr 452 . . . . . 6  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  M  Isom  _E  ,  R  ( dom  M ,  X ) )
68 isocnv 6050 . . . . . 6  |-  ( M 
Isom  _E  ,  R  ( dom  M ,  X
)  ->  `' M  Isom  R ,  _E  ( X ,  dom  M ) )
6967, 68syl 16 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  `' M  Isom  R ,  _E  ( X ,  dom  M ) )
7039adantrr 698 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  C  e.  X
)
7131simpld 446 . . . . . . . . . 10  |-  ( ph  ->  ( X  C_  A  /\  R  C_  ( X  X.  X ) ) )
7271simprd 450 . . . . . . . . 9  |-  ( ph  ->  R  C_  ( X  X.  X ) )
7372ssbrd 4253 . . . . . . . 8  |-  ( ph  ->  ( D R ( M `  B )  ->  D ( X  X.  X ) ( M `  B ) ) )
7473imp 419 . . . . . . 7  |-  ( (
ph  /\  D R
( M `  B
) )  ->  D
( X  X.  X
) ( M `  B ) )
75 brxp 4909 . . . . . . . 8  |-  ( D ( X  X.  X
) ( M `  B )  <->  ( D  e.  X  /\  ( M `  B )  e.  X ) )
7675simplbi 447 . . . . . . 7  |-  ( D ( X  X.  X
) ( M `  B )  ->  D  e.  X )
7774, 76syl 16 . . . . . 6  |-  ( (
ph  /\  D R
( M `  B
) )  ->  D  e.  X )
7877adantrl 697 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  D  e.  X
)
79 isorel 6046 . . . . 5  |-  ( ( `' M  Isom  R ,  _E  ( X ,  dom  M )  /\  ( C  e.  X  /\  D  e.  X ) )  -> 
( C R D  <-> 
( `' M `  C )  _E  ( `' M `  D ) ) )
8069, 70, 78, 79syl12anc 1182 . . . 4  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  ( C R D  <->  ( `' M `  C )  _E  ( `' M `  D ) ) )
8113adantr 452 . . . . . 6  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  N  Isom  _E  ,  S  ( dom  N ,  Y ) )
82 isocnv 6050 . . . . . 6  |-  ( N 
Isom  _E  ,  S  ( dom  N ,  Y
)  ->  `' N  Isom  S ,  _E  ( Y ,  dom  N ) )
8381, 82syl 16 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  `' N  Isom  S ,  _E  ( Y ,  dom  N ) )
8424adantrr 698 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  C  e.  Y
)
8563simp2d 970 . . . . . 6  |-  ( (
ph  /\  D R
( M `  B
) )  ->  D  e.  Y )
8685adantrl 697 . . . . 5  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  D  e.  Y
)
87 isorel 6046 . . . . 5  |-  ( ( `' N  Isom  S ,  _E  ( Y ,  dom  N )  /\  ( C  e.  Y  /\  D  e.  Y ) )  -> 
( C S D  <-> 
( `' N `  C )  _E  ( `' N `  D ) ) )
8883, 84, 86, 87syl12anc 1182 . . . 4  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  ( C S D  <->  ( `' N `  C )  _E  ( `' N `  D ) ) )
8966, 80, 883bitr4d 277 . . 3  |-  ( (
ph  /\  ( C R ( M `  B )  /\  D R ( M `  B ) ) )  ->  ( C R D  <->  C S D ) )
9089expr 599 . 2  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( D R ( M `  B )  ->  ( C R D  <->  C S D ) ) )
9161, 90jca 519 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956   [.wsbc 3161    i^i cin 3319    C_ wss 3320   {csn 3814   class class class wbr 4212   {copab 4265    _E cep 4492    We wwe 4540    X. cxp 4876   `'ccnv 4877   dom cdm 4878    |` cres 4880   "cima 4881   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454    Isom wiso 5455  (class class class)co 6081  OrdIsocoi 7478
This theorem is referenced by:  fpwwe2lem8  8512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-riota 6549  df-recs 6633  df-oi 7479
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