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Theorem fpwwe2lem6 9030
Description: Lemma for fpwwe2 9038. (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 9027 . . . . . . . 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 210 . . . . . . 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 459 . . . . . 6  |-  ( ph  ->  ( X  C_  A  /\  R  C_  ( X  X.  X ) ) )
76simprd 463 . . . . 5  |-  ( ph  ->  R  C_  ( X  X.  X ) )
87ssbrd 4497 . . . 4  |-  ( ph  ->  ( C R ( M `  B )  ->  C ( X  X.  X ) ( M `  B ) ) )
9 brxp 5039 . . . . 5  |-  ( C ( X  X.  X
) ( M `  B )  <->  ( C  e.  X  /\  ( M `  B )  e.  X ) )
109simplbi 460 . . . 4  |-  ( C ( X  X.  X
) ( M `  B )  ->  C  e.  X )
118, 10syl6 33 . . 3  |-  ( ph  ->  ( C R ( M `  B )  ->  C  e.  X
) )
1211imp 429 . 2  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  X )
13 imassrn 5358 . . . 4  |-  ( N
" B )  C_  ran  N
14 fpwwe2lem9.y . . . . . . . . 9  |-  ( ph  ->  Y W S )
152relopabi 5137 . . . . . . . . . 10  |-  Rel  W
1615brrelexi 5049 . . . . . . . . 9  |-  ( Y W S  ->  Y  e.  _V )
1714, 16syl 16 . . . . . . . 8  |-  ( ph  ->  Y  e.  _V )
182, 3fpwwe2lem2 9027 . . . . . . . . . . 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 210 . . . . . . . . . 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 463 . . . . . . . . 9  |-  ( ph  ->  ( S  We  Y  /\  A. y  e.  Y  [. ( `' S " { y } )  /  u ]. (
u F ( S  i^i  ( u  X.  u ) ) )  =  y ) )
2120simpld 459 . . . . . . . 8  |-  ( ph  ->  S  We  Y )
22 fpwwe2lem9.n . . . . . . . . 9  |-  N  = OrdIso
( S ,  Y
)
2322oiiso 7980 . . . . . . . 8  |-  ( ( Y  e.  _V  /\  S  We  Y )  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
2417, 21, 23syl2anc 661 . . . . . . 7  |-  ( ph  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
2524adantr 465 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  N  Isom  _E  ,  S  ( dom  N ,  Y
) )
26 isof1o 6222 . . . . . 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 5829 . . . . 5  |-  ( N : dom  N -1-1-onto-> Y  ->  N : dom  N -onto-> Y
)
29 forn 5804 . . . . 5  |-  ( N : dom  N -onto-> Y  ->  ran  N  =  Y )
3027, 28, 293syl 20 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ran  N  =  Y )
3113, 30syl5sseq 3547 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( N " B )  C_  Y )
3215brrelexi 5049 . . . . . . . . . . . . . 14  |-  ( X W R  ->  X  e.  _V )
331, 32syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  X  e.  _V )
345simprd 463 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( R  We  X  /\  A. y  e.  X  [. ( `' R " { y } )  /  u ]. (
u F ( R  i^i  ( u  X.  u ) ) )  =  y ) )
3534simpld 459 . . . . . . . . . . . . 13  |-  ( ph  ->  R  We  X )
36 fpwwe2lem9.m . . . . . . . . . . . . . 14  |-  M  = OrdIso
( R ,  X
)
3736oiiso 7980 . . . . . . . . . . . . 13  |-  ( ( X  e.  _V  /\  R  We  X )  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
3833, 35, 37syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
3938adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  C R
( M `  B
) )  ->  M  Isom  _E  ,  R  ( dom  M ,  X
) )
40 isof1o 6222 . . . . . . . . . . 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 6184 . . . . . . . . . 10  |-  ( ( M : dom  M -1-1-onto-> X  /\  C  e.  X
)  ->  ( M `  ( `' M `  C ) )  =  C )
4341, 12, 42syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M `  ( `' M `  C )
)  =  C )
44 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C R ( M `  B ) )
4543, 44eqbrtrd 4476 . . . . . . . 8  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M `  ( `' M `  C )
) R ( M `
 B ) )
46 f1ocnv 5834 . . . . . . . . . . 11  |-  ( M : dom  M -1-1-onto-> X  ->  `' M : X -1-1-onto-> dom  M
)
47 f1of 5822 . . . . . . . . . . 11  |-  ( `' M : X -1-1-onto-> dom  M  ->  `' M : X --> dom  M
)
4841, 46, 473syl 20 . . . . . . . . . 10  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' M : X --> dom  M
)
4948, 12ffvelrnd 6033 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  e.  dom  M )
50 fpwwe2lem7.1 . . . . . . . . . 10  |-  ( ph  ->  B  e.  dom  M
)
5150adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  C R
( M `  B
) )  ->  B  e.  dom  M )
52 isorel 6223 . . . . . . . . 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 1226 . . . . . . . 8  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' M `  C )  _E  B  <->  ( M `  ( `' M `  C ) ) R ( M `
 B ) ) )
5445, 53mpbird 232 . . . . . . 7  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  _E  B )
55 epelg 4801 . . . . . . . 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 210 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  e.  B )
58 ffn 5737 . . . . . . 7  |-  ( `' M : X --> dom  M  ->  `' M  Fn  X
)
59 elpreima 6008 . . . . . . 7  |-  ( `' M  Fn  X  -> 
( C  e.  ( `' `' M " B )  <-> 
( C  e.  X  /\  ( `' M `  C )  e.  B
) ) )
6048, 58, 593syl 20 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( C  e.  ( `' `' M " B )  <-> 
( C  e.  X  /\  ( `' M `  C )  e.  B
) ) )
6112, 57, 60mpbir2and 922 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  ( `' `' M " B ) )
62 imacnvcnv 5478 . . . . 5  |-  ( `' `' M " B )  =  ( M " B )
6361, 62syl6eleq 2555 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  ( M " B
) )
64 fpwwe2lem7.3 . . . . . . 7  |-  ( ph  ->  ( M  |`  B )  =  ( N  |`  B ) )
6564adantr 465 . . . . . 6  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M  |`  B )  =  ( N  |`  B ) )
6665rneqd 5240 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ran  ( M  |`  B )  =  ran  ( N  |`  B ) )
67 df-ima 5021 . . . . 5  |-  ( M
" B )  =  ran  ( M  |`  B )
68 df-ima 5021 . . . . 5  |-  ( N
" B )  =  ran  ( N  |`  B )
6966, 67, 683eqtr4g 2523 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( M " B )  =  ( N " B
) )
7063, 69eleqtrd 2547 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  ( N " B
) )
7131, 70sseldd 3500 . 2  |-  ( (
ph  /\  C R
( M `  B
) )  ->  C  e.  Y )
7265cnveqd 5188 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' ( M  |`  B )  =  `' ( N  |`  B ) )
73 dff1o3 5828 . . . . . . 7  |-  ( M : dom  M -1-1-onto-> X  <->  ( M : dom  M -onto-> X  /\  Fun  `' M ) )
7473simprbi 464 . . . . . 6  |-  ( M : dom  M -1-1-onto-> X  ->  Fun  `' M )
75 funcnvres 5663 . . . . . 6  |-  ( Fun  `' M  ->  `' ( M  |`  B )  =  ( `' M  |`  ( M " B
) ) )
7641, 74, 753syl 20 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' ( M  |`  B )  =  ( `' M  |`  ( M " B
) ) )
77 dff1o3 5828 . . . . . . 7  |-  ( N : dom  N -1-1-onto-> Y  <->  ( N : dom  N -onto-> Y  /\  Fun  `' N ) )
7877simprbi 464 . . . . . 6  |-  ( N : dom  N -1-1-onto-> Y  ->  Fun  `' N )
79 funcnvres 5663 . . . . . 6  |-  ( Fun  `' N  ->  `' ( N  |`  B )  =  ( `' N  |`  ( N " B
) ) )
8027, 78, 793syl 20 . . . . 5  |-  ( (
ph  /\  C R
( M `  B
) )  ->  `' ( N  |`  B )  =  ( `' N  |`  ( N " B
) ) )
8172, 76, 803eqtr3d 2506 . . . 4  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M  |`  ( M
" B ) )  =  ( `' N  |`  ( N " B
) ) )
8281fveq1d 5874 . . 3  |-  ( (
ph  /\  C R
( M `  B
) )  ->  (
( `' M  |`  ( M " B ) ) `  C )  =  ( ( `' N  |`  ( N " B ) ) `  C ) )
83 fvres 5886 . . . 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 5886 . . . 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 2506 . 2  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( `' M `  C )  =  ( `' N `  C ) )
8812, 71, 873jca 1176 1  |-  ( (
ph  /\  C R
( M `  B
) )  ->  ( C  e.  X  /\  C  e.  Y  /\  ( `' M `  C )  =  ( `' N `  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   [.wsbc 3327    i^i cin 3470    C_ wss 3471   {csn 4032   class class class wbr 4456   {copab 4514    _E cep 4798    We wwe 4846    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011   Fun wfun 5588    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595  (class class class)co 6296  OrdIsocoi 7952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-recs 7060  df-oi 7953
This theorem is referenced by:  fpwwe2lem7  9031
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