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Theorem fpwwe2lem6 8470
Description: Lemma for fpwwe2 8478. (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 8467 . . . . . . . 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 202 . . . . . . 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 446 . . . . . 6  |-  ( ph  ->  ( X  C_  A  /\  R  C_  ( X  X.  X ) ) )
76simprd 450 . . . . 5  |-  ( ph  ->  R  C_  ( X  X.  X ) )
87ssbrd 4217 . . . 4  |-  ( ph  ->  ( C R ( M `  B )  ->  C ( X  X.  X ) ( M `  B ) ) )
9 brxp 4872 . . . . 5  |-  ( C ( X  X.  X
) ( M `  B )  <->  ( C  e.  X  /\  ( M `  B )  e.  X ) )
109simplbi 447 . . . 4  |-  ( C ( X  X.  X
) ( M `  B )  ->  C  e.  X )
118, 10syl6 31 . . 3  |-  ( ph  ->  ( C R ( M `  B )  ->  C  e.  X
) )
1211imp 419 . 2  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  X )
13 imassrn 5179 . . . 4  |-  ( N
" B )  C_  ran  N
14 fpwwe2lem9.y . . . . . . . . 9  |-  ( ph  ->  Y W S )
152relopabi 4963 . . . . . . . . . 10  |-  Rel  W
1615brrelexi 4881 . . . . . . . . 9  |-  ( Y W S  ->  Y  e.  _V )
1714, 16syl 16 . . . . . . . 8  |-  ( ph  ->  Y  e.  _V )
182, 3fpwwe2lem2 8467 . . . . . . . . . . 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 202 . . . . . . . . . 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 450 . . . . . . . . 9  |-  ( ph  ->  ( S  We  Y  /\  A. y  e.  Y  [. ( `' S " { y } )  /  u ]. (
u F ( S  i^i  ( u  X.  u ) ) )  =  y ) )
2120simpld 446 . . . . . . . 8  |-  ( ph  ->  S  We  Y )
22 fpwwe2lem9.n . . . . . . . . 9  |-  N  = OrdIso
( S ,  Y
)
2322oiiso 7466 . . . . . . . 8  |-  ( ( Y  e.  _V  /\  S  We  Y )  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
2417, 21, 23syl2anc 643 . . . . . . 7  |-  ( ph  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
2524adantr 452 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
26 isof1o 6008 . . . . . 6  |-  ( N 
Isom  _E  ,  S  ( dom  N ,  Y
)  ->  N : dom  N -1-1-onto-> Y )
2725, 26syl 16 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  N : dom  N -1-1-onto-> Y )
28 f1ofo 5644 . . . . 5  |-  ( N : dom  N -1-1-onto-> Y  ->  N : dom  N -onto-> Y
)
29 forn 5619 . . . . 5  |-  ( N : dom  N -onto-> Y  ->  ran  N  =  Y )
3027, 28, 293syl 19 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ran  N  =  Y )
3113, 30syl5sseq 3360 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( N " B )  C_  Y )
3215brrelexi 4881 . . . . . . . . . . . . . 14  |-  ( X W R  ->  X  e.  _V )
331, 32syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  X  e.  _V )
345simprd 450 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( R  We  X  /\  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
3534simpld 446 . . . . . . . . . . . . 13  |-  ( ph  ->  R  We  X )
36 fpwwe2lem9.m . . . . . . . . . . . . . 14  |-  M  = OrdIso
( R ,  X
)
3736oiiso 7466 . . . . . . . . . . . . 13  |-  ( ( X  e.  _V  /\  R  We  X )  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
3833, 35, 37syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
3938adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  C R
( M `  B
) )  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
40 isof1o 6008 . . . . . . . . . . 11  |-  ( M 
Isom  _E  ,  R  ( dom  M ,  X
)  ->  M : dom  M -1-1-onto-> X )
4139, 40syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  C R
( M `  B
) )  ->  M : dom  M -1-1-onto-> X )
42 f1ocnvfv2 5978 . . . . . . . . . 10  |-  ( ( M : dom  M -1-1-onto-> X  /\  C  e.  X
)  ->  ( M `  ( `' M `  C ) )  =  C )
4341, 12, 42syl2anc 643 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M `  ( `' M `  C )
)  =  C )
44 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C R ( M `  B ) )
4543, 44eqbrtrd 4196 . . . . . . . 8  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M `  ( `' M `  C )
) R ( M `
 B ) )
46 f1ocnv 5650 . . . . . . . . . . 11  |-  ( M : dom  M -1-1-onto-> X  ->  `' M : X -1-1-onto-> dom  M
)
47 f1of 5637 . . . . . . . . . . 11  |-  ( `' M : X -1-1-onto-> dom  M  ->  `' M : X --> dom  M
)
4841, 46, 473syl 19 . . . . . . . . . 10  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' M : X --> dom  M
)
4948, 12ffvelrnd 5834 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  e.  dom  M )
50 fpwwe2lem7.1 . . . . . . . . . 10  |-  ( ph  ->  B  e.  dom  M
)
5150adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  B  e.  dom  M )
52 isorel 6009 . . . . . . . . 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
) ) )
5339, 49, 51, 52syl12anc 1182 . . . . . . . 8  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' M `  C )  _E  B  <->  ( M `  ( `' M `  C ) ) R ( M `
 B ) ) )
5445, 53mpbird 224 . . . . . . 7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  _E  B )
55 epelg 4459 . . . . . . . 8  |-  ( B  e.  dom  M  -> 
( ( `' M `  C )  _E  B  <->  ( `' M `  C )  e.  B ) )
5651, 55syl 16 . . . . . . 7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' M `  C )  _E  B  <->  ( `' M `  C )  e.  B ) )
5754, 56mpbid 202 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  e.  B )
58 ffn 5554 . . . . . . 7  |-  ( `' M : X --> dom  M  ->  `' M  Fn  X
)
59 elpreima 5813 . . . . . . 7  |-  ( `' M  Fn  X  -> 
( C  e.  ( `' `' M " B )  <-> 
( C  e.  X  /\  ( `' M `  C )  e.  B
) ) )
6048, 58, 593syl 19 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( C  e.  ( `' `' M " B )  <-> 
( C  e.  X  /\  ( `' M `  C )  e.  B
) ) )
6112, 57, 60mpbir2and 889 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  ( `' `' M " B ) )
62 imacnvcnv 5297 . . . . 5  |-  ( `' `' M " B )  =  ( M " B )
6361, 62syl6eleq 2498 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  ( M " B
) )
64 fpwwe2lem7.3 . . . . . . 7  |-  ( ph  ->  ( M  |`  B )  =  ( N  |`  B ) )
6564adantr 452 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M  |`  B )  =  ( N  |`  B ) )
6665rneqd 5060 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ran  ( M  |`  B )  =  ran  ( N  |`  B ) )
67 df-ima 4854 . . . . 5  |-  ( M
" B )  =  ran  ( M  |`  B )
68 df-ima 4854 . . . . 5  |-  ( N
" B )  =  ran  ( N  |`  B )
6966, 67, 683eqtr4g 2465 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M " B )  =  ( N " B
) )
7063, 69eleqtrd 2484 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  ( N " B
) )
7131, 70sseldd 3313 . 2  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  Y )
7265cnveqd 5011 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' ( M  |`  B )  =  `' ( N  |`  B ) )
73 dff1o3 5643 . . . . . . 7  |-  ( M : dom  M -1-1-onto-> X  <->  ( M : dom  M -onto-> X  /\  Fun  `' M ) )
7473simprbi 451 . . . . . 6  |-  ( M : dom  M -1-1-onto-> X  ->  Fun  `' M )
75 funcnvres 5485 . . . . . 6  |-  ( Fun  `' M  ->  `' ( M  |`  B )  =  ( `' M  |`  ( M " B
) ) )
7641, 74, 753syl 19 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' ( M  |`  B )  =  ( `' M  |`  ( M " B
) ) )
77 dff1o3 5643 . . . . . . 7  |-  ( N : dom  N -1-1-onto-> Y  <->  ( N : dom  N -onto-> Y  /\  Fun  `' N ) )
7877simprbi 451 . . . . . 6  |-  ( N : dom  N -1-1-onto-> Y  ->  Fun  `' N )
79 funcnvres 5485 . . . . . 6  |-  ( Fun  `' N  ->  `' ( N  |`  B )  =  ( `' N  |`  ( N " B
) ) )
8027, 78, 793syl 19 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' ( N  |`  B )  =  ( `' N  |`  ( N " B
) ) )
8172, 76, 803eqtr3d 2448 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M  |`  ( M
" B ) )  =  ( `' N  |`  ( N " B
) ) )
8281fveq1d 5693 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' M  |`  ( M " B ) ) `  C )  =  ( ( `' N  |`  ( N " B ) ) `  C ) )
83 fvres 5708 . . . 4  |-  ( C  e.  ( M " B )  ->  (
( `' M  |`  ( M " B ) ) `  C )  =  ( `' M `  C ) )
8463, 83syl 16 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' M  |`  ( M " B ) ) `  C )  =  ( `' M `  C ) )
85 fvres 5708 . . . 4  |-  ( C  e.  ( N " B )  ->  (
( `' N  |`  ( N " B ) ) `  C )  =  ( `' N `  C ) )
8670, 85syl 16 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' N  |`  ( N " B ) ) `  C )  =  ( `' N `  C ) )
8782, 84, 863eqtr3d 2448 . 2  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  =  ( `' N `  C ) )
8812, 71, 873jca 1134 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2670   _Vcvv 2920   [.wsbc 3125    i^i cin 3283    C_ wss 3284   {csn 3778   class class class wbr 4176   {copab 4229    _E cep 4456    We wwe 4504    X. cxp 4839   `'ccnv 4840   dom cdm 4841   ran crn 4842    |` cres 4843   "cima 4844   Fun wfun 5411    Fn wfn 5412   -->wf 5413   -onto->wfo 5415   -1-1-onto->wf1o 5416   ` cfv 5417    Isom wiso 5418  (class class class)co 6044  OrdIsocoi 7438
This theorem is referenced by:  fpwwe2lem7  8471
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-riota 6512  df-recs 6596  df-oi 7439
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