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Theorem fpwwe2lem6 8257
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
fpwwe2lem6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( C  e.  X  /\  C  e.  Y  /\  ( `' M `  C )  =  ( `' N `  C ) ) )
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)

Proof of Theorem fpwwe2lem6
StepHypRef Expression
1 fpwwe2lem9.x . . . . . . . 8  |-  ( ph  ->  X W R )
2 fpwwe2.1 . . . . . . . . 9  |-  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 ) ) }
3 fpwwe2.2 . . . . . . . . 9  |-  ( ph  ->  A  e.  _V )
42, 3fpwwe2lem2 8254 . . . . . . . 8  |-  ( 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 ) ) ) )
51, 4mpbid 201 . . . . . . 7  |-  ( 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 ) ) )
65simpld 445 . . . . . 6  |-  ( ph  ->  ( X  C_  A  /\  R  C_  ( X  X.  X ) ) )
76simprd 449 . . . . 5  |-  ( ph  ->  R  C_  ( X  X.  X ) )
87ssbrd 4064 . . . 4  |-  ( ph  ->  ( C R ( M `  B )  ->  C ( X  X.  X ) ( M `  B ) ) )
9 brxp 4720 . . . . 5  |-  ( C ( X  X.  X
) ( M `  B )  <->  ( C  e.  X  /\  ( M `  B )  e.  X ) )
109simplbi 446 . . . 4  |-  ( C ( X  X.  X
) ( M `  B )  ->  C  e.  X )
118, 10syl6 29 . . 3  |-  ( ph  ->  ( C R ( M `  B )  ->  C  e.  X
) )
1211imp 418 . 2  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  X )
13 imassrn 5025 . . . 4  |-  ( N
" B )  C_  ran  N
14 fpwwe2lem9.y . . . . . . . . 9  |-  ( ph  ->  Y W S )
152relopabi 4811 . . . . . . . . . 10  |-  Rel  W
1615brrelexi 4729 . . . . . . . . 9  |-  ( Y W S  ->  Y  e.  _V )
1714, 16syl 15 . . . . . . . 8  |-  ( ph  ->  Y  e.  _V )
182, 3fpwwe2lem2 8254 . . . . . . . . . . 11  |-  ( 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 ) ) ) )
1914, 18mpbid 201 . . . . . . . . . 10  |-  ( 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 ) ) )
2019simprd 449 . . . . . . . . 9  |-  ( ph  ->  ( S  We  Y  /\  A. y  e.  Y  [. ( `' S " { y } )  /  u ]. (
u F ( S  i^i  ( u  X.  u ) ) )  =  y ) )
2120simpld 445 . . . . . . . 8  |-  ( ph  ->  S  We  Y )
22 fpwwe2lem9.n . . . . . . . . 9  |-  N  = OrdIso
( S ,  Y
)
2322oiiso 7252 . . . . . . . 8  |-  ( ( Y  e.  _V  /\  S  We  Y )  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
2417, 21, 23syl2anc 642 . . . . . . 7  |-  ( ph  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
2524adantr 451 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
26 isof1o 5822 . . . . . 6  |-  ( N 
Isom  _E  ,  S  ( dom  N ,  Y
)  ->  N : dom  N -1-1-onto-> Y )
2725, 26syl 15 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  N : dom  N -1-1-onto-> Y )
28 f1ofo 5479 . . . . 5  |-  ( N : dom  N -1-1-onto-> Y  ->  N : dom  N -onto-> Y
)
29 forn 5454 . . . . 5  |-  ( N : dom  N -onto-> Y  ->  ran  N  =  Y )
3027, 28, 293syl 18 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ran  N  =  Y )
3113, 30syl5sseq 3226 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( N " B )  C_  Y )
3215brrelexi 4729 . . . . . . . . . . . . . 14  |-  ( X W R  ->  X  e.  _V )
331, 32syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  X  e.  _V )
345simprd 449 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( R  We  X  /\  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
3534simpld 445 . . . . . . . . . . . . 13  |-  ( ph  ->  R  We  X )
36 fpwwe2lem9.m . . . . . . . . . . . . . 14  |-  M  = OrdIso
( R ,  X
)
3736oiiso 7252 . . . . . . . . . . . . 13  |-  ( ( X  e.  _V  /\  R  We  X )  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
3833, 35, 37syl2anc 642 . . . . . . . . . . . 12  |-  ( ph  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
3938adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  C R
( M `  B
) )  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
40 isof1o 5822 . . . . . . . . . . 11  |-  ( M 
Isom  _E  ,  R  ( dom  M ,  X
)  ->  M : dom  M -1-1-onto-> X )
4139, 40syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  C R
( M `  B
) )  ->  M : dom  M -1-1-onto-> X )
42 f1ocnvfv2 5793 . . . . . . . . . 10  |-  ( ( M : dom  M -1-1-onto-> X  /\  C  e.  X
)  ->  ( M `  ( `' M `  C ) )  =  C )
4341, 12, 42syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M `  ( `' M `  C )
)  =  C )
44 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C R ( M `  B ) )
4543, 44eqbrtrd 4043 . . . . . . . 8  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M `  ( `' M `  C )
) R ( M `
 B ) )
46 f1ocnv 5485 . . . . . . . . . . 11  |-  ( M : dom  M -1-1-onto-> X  ->  `' M : X -1-1-onto-> dom  M
)
47 f1of 5472 . . . . . . . . . . 11  |-  ( `' M : X -1-1-onto-> dom  M  ->  `' M : X --> dom  M
)
4841, 46, 473syl 18 . . . . . . . . . 10  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' M : X --> dom  M
)
49 ffvelrn 5663 . . . . . . . . . 10  |-  ( ( `' M : X --> dom  M  /\  C  e.  X
)  ->  ( `' M `  C )  e.  dom  M )
5048, 12, 49syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  e.  dom  M )
51 fpwwe2lem7.1 . . . . . . . . . 10  |-  ( ph  ->  B  e.  dom  M
)
5251adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  B  e.  dom  M )
53 isorel 5823 . . . . . . . . 9  |-  ( ( 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
) ) )
5439, 50, 52, 53syl12anc 1180 . . . . . . . 8  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' M `  C )  _E  B  <->  ( M `  ( `' M `  C ) ) R ( M `
 B ) ) )
5545, 54mpbird 223 . . . . . . 7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  _E  B )
56 epelg 4306 . . . . . . . 8  |-  ( B  e.  dom  M  -> 
( ( `' M `  C )  _E  B  <->  ( `' M `  C )  e.  B ) )
5752, 56syl 15 . . . . . . 7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' M `  C )  _E  B  <->  ( `' M `  C )  e.  B ) )
5855, 57mpbid 201 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  e.  B )
59 ffn 5389 . . . . . . 7  |-  ( `' M : X --> dom  M  ->  `' M  Fn  X
)
60 elpreima 5645 . . . . . . 7  |-  ( `' M  Fn  X  -> 
( C  e.  ( `' `' M " B )  <-> 
( C  e.  X  /\  ( `' M `  C )  e.  B
) ) )
6148, 59, 603syl 18 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( C  e.  ( `' `' M " B )  <-> 
( C  e.  X  /\  ( `' M `  C )  e.  B
) ) )
6212, 58, 61mpbir2and 888 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  ( `' `' M " B ) )
63 imacnvcnv 5137 . . . . 5  |-  ( `' `' M " B )  =  ( M " B )
6462, 63syl6eleq 2373 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  ( M " B
) )
65 fpwwe2lem7.3 . . . . . . 7  |-  ( ph  ->  ( M  |`  B )  =  ( N  |`  B ) )
6665adantr 451 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M  |`  B )  =  ( N  |`  B ) )
6766rneqd 4906 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ran  ( M  |`  B )  =  ran  ( N  |`  B ) )
68 df-ima 4702 . . . . 5  |-  ( M
" B )  =  ran  ( M  |`  B )
69 df-ima 4702 . . . . 5  |-  ( N
" B )  =  ran  ( N  |`  B )
7067, 68, 693eqtr4g 2340 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M " B )  =  ( N " B
) )
7164, 70eleqtrd 2359 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  ( N " B
) )
7231, 71sseldd 3181 . 2  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  Y )
7366cnveqd 4857 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' ( M  |`  B )  =  `' ( N  |`  B ) )
74 dff1o3 5478 . . . . . . 7  |-  ( M : dom  M -1-1-onto-> X  <->  ( M : dom  M -onto-> X  /\  Fun  `' M ) )
7574simprbi 450 . . . . . 6  |-  ( M : dom  M -1-1-onto-> X  ->  Fun  `' M )
76 funcnvres 5321 . . . . . 6  |-  ( Fun  `' M  ->  `' ( M  |`  B )  =  ( `' M  |`  ( M " B
) ) )
7741, 75, 763syl 18 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' ( M  |`  B )  =  ( `' M  |`  ( M " B
) ) )
78 dff1o3 5478 . . . . . . 7  |-  ( N : dom  N -1-1-onto-> Y  <->  ( N : dom  N -onto-> Y  /\  Fun  `' N ) )
7978simprbi 450 . . . . . 6  |-  ( N : dom  N -1-1-onto-> Y  ->  Fun  `' N )
80 funcnvres 5321 . . . . . 6  |-  ( Fun  `' N  ->  `' ( N  |`  B )  =  ( `' N  |`  ( N " B
) ) )
8127, 79, 803syl 18 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' ( N  |`  B )  =  ( `' N  |`  ( N " B
) ) )
8273, 77, 813eqtr3d 2323 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M  |`  ( M
" B ) )  =  ( `' N  |`  ( N " B
) ) )
8382fveq1d 5527 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' M  |`  ( M " B ) ) `  C )  =  ( ( `' N  |`  ( N " B ) ) `  C ) )
84 fvres 5542 . . . 4  |-  ( C  e.  ( M " B )  ->  (
( `' M  |`  ( M " B ) ) `  C )  =  ( `' M `  C ) )
8564, 84syl 15 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' M  |`  ( M " B ) ) `  C )  =  ( `' M `  C ) )
86 fvres 5542 . . . 4  |-  ( C  e.  ( N " B )  ->  (
( `' N  |`  ( N " B ) ) `  C )  =  ( `' N `  C ) )
8771, 86syl 15 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' N  |`  ( N " B ) ) `  C )  =  ( `' N `  C ) )
8883, 85, 873eqtr3d 2323 . 2  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  =  ( `' N `  C ) )
8912, 72, 883jca 1132 1  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( C  e.  X  /\  C  e.  Y  /\  ( `' M `  C )  =  ( `' N `  C ) ) )
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   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255    Isom wiso 5256  (class class class)co 5858  OrdIsocoi 7224
This theorem is referenced by:  fpwwe2lem7  8258
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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