MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fparlem4 Structured version   Unicode version

Theorem fparlem4 6910
Description: Lemma for fpar 6911. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fparlem4  |-  ( G  Fn  B  ->  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  =  U_ y  e.  B  ( ( _V  X.  { y } )  X.  ( _V 
X.  { ( G `
 y ) } ) ) )
Distinct variable groups:    y, B    y, G

Proof of Theorem fparlem4
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 coiun 5365 . 2  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  U_ y  e.  B  ( ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  X.  ( G
" { y } ) ) )  = 
U_ y  e.  B  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  X.  ( G
" { y } ) ) )
2 inss1 3688 . . . . 5  |-  ( dom 
G  i^i  ran  ( 2nd  |`  ( _V  X.  _V ) ) )  C_  dom  G
3 fndm 5693 . . . . 5  |-  ( G  Fn  B  ->  dom  G  =  B )
42, 3syl5sseq 3518 . . . 4  |-  ( G  Fn  B  ->  ( dom  G  i^i  ran  ( 2nd  |`  ( _V  X.  _V ) ) )  C_  B )
5 dfco2a 5355 . . . 4  |-  ( ( dom  G  i^i  ran  ( 2nd  |`  ( _V  X.  _V ) ) ) 
C_  B  ->  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  = 
U_ y  e.  B  ( ( `' ( 2nd  |`  ( _V  X.  _V ) ) " { y } )  X.  ( G " { y } ) ) )
64, 5syl 17 . . 3  |-  ( G  Fn  B  ->  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  = 
U_ y  e.  B  ( ( `' ( 2nd  |`  ( _V  X.  _V ) ) " { y } )  X.  ( G " { y } ) ) )
76coeq2d 5017 . 2  |-  ( G  Fn  B  ->  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  =  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  U_ y  e.  B  ( ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  X.  ( G
" { y } ) ) ) )
8 inss1 3688 . . . . . . . . 9  |-  ( dom  ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  i^i  ran  ( 2nd  |`  ( _V  X.  _V ) ) )  C_  dom  ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )
9 dmxpss 5288 . . . . . . . . 9  |-  dom  ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  C_  { ( G `  y
) }
108, 9sstri 3479 . . . . . . . 8  |-  ( dom  ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  i^i  ran  ( 2nd  |`  ( _V  X.  _V ) ) )  C_  { ( G `  y
) }
11 dfco2a 5355 . . . . . . . 8  |-  ( ( dom  ( { ( G `  y ) }  X.  ( _V 
X.  { y } ) )  i^i  ran  ( 2nd  |`  ( _V  X.  _V ) ) ) 
C_  { ( G `
 y ) }  ->  ( ( { ( G `  y
) }  X.  ( _V  X.  { y } ) )  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  = 
U_ x  e.  {
( G `  y
) }  ( ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { x }
)  X.  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) ) " { x } ) ) )
1210, 11ax-mp 5 . . . . . . 7  |-  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  U_ x  e.  { ( G `  y ) }  (
( `' ( 2nd  |`  ( _V  X.  _V ) ) " {
x } )  X.  ( ( { ( G `  y ) }  X.  ( _V 
X.  { y } ) ) " {
x } ) )
13 fvex 5891 . . . . . . . 8  |-  ( G `
 y )  e. 
_V
14 fparlem2 6908 . . . . . . . . . 10  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { x }
)  =  ( _V 
X.  { x }
)
15 sneq 4012 . . . . . . . . . . 11  |-  ( x  =  ( G `  y )  ->  { x }  =  { ( G `  y ) } )
1615xpeq2d 4878 . . . . . . . . . 10  |-  ( x  =  ( G `  y )  ->  ( _V  X.  { x }
)  =  ( _V 
X.  { ( G `
 y ) } ) )
1714, 16syl5eq 2482 . . . . . . . . 9  |-  ( x  =  ( G `  y )  ->  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { x }
)  =  ( _V 
X.  { ( G `
 y ) } ) )
1815imaeq2d 5188 . . . . . . . . . 10  |-  ( x  =  ( G `  y )  ->  (
( { ( G `
 y ) }  X.  ( _V  X.  { y } ) ) " { x } )  =  ( ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) ) " { ( G `  y ) } ) )
19 df-ima 4867 . . . . . . . . . . 11  |-  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) ) " { ( G `  y ) } )  =  ran  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  |`  { ( G `  y ) } )
20 ssid 3489 . . . . . . . . . . . . . 14  |-  { ( G `  y ) }  C_  { ( G `  y ) }
21 xpssres 5159 . . . . . . . . . . . . . 14  |-  ( { ( G `  y
) }  C_  { ( G `  y ) }  ->  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  |`  { ( G `  y ) } )  =  ( { ( G `  y ) }  X.  ( _V 
X.  { y } ) ) )
2220, 21ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  |`  { ( G `  y ) } )  =  ( { ( G `  y ) }  X.  ( _V 
X.  { y } ) )
2322rneqi 5081 . . . . . . . . . . . 12  |-  ran  (
( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  |`  { ( G `  y ) } )  =  ran  ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )
2413snnz 4121 . . . . . . . . . . . . 13  |-  { ( G `  y ) }  =/=  (/)
25 rnxp 5287 . . . . . . . . . . . . 13  |-  ( { ( G `  y
) }  =/=  (/)  ->  ran  ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  =  ( _V  X.  {
y } ) )
2624, 25ax-mp 5 . . . . . . . . . . . 12  |-  ran  ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  =  ( _V  X.  {
y } )
2723, 26eqtri 2458 . . . . . . . . . . 11  |-  ran  (
( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  |`  { ( G `  y ) } )  =  ( _V  X.  { y } )
2819, 27eqtri 2458 . . . . . . . . . 10  |-  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) ) " { ( G `  y ) } )  =  ( _V  X.  { y } )
2918, 28syl6eq 2486 . . . . . . . . 9  |-  ( x  =  ( G `  y )  ->  (
( { ( G `
 y ) }  X.  ( _V  X.  { y } ) ) " { x } )  =  ( _V  X.  { y } ) )
3017, 29xpeq12d 4879 . . . . . . . 8  |-  ( x  =  ( G `  y )  ->  (
( `' ( 2nd  |`  ( _V  X.  _V ) ) " {
x } )  X.  ( ( { ( G `  y ) }  X.  ( _V 
X.  { y } ) ) " {
x } ) )  =  ( ( _V 
X.  { ( G `
 y ) } )  X.  ( _V 
X.  { y } ) ) )
3113, 30iunxsn 4385 . . . . . . 7  |-  U_ x  e.  { ( G `  y ) }  (
( `' ( 2nd  |`  ( _V  X.  _V ) ) " {
x } )  X.  ( ( { ( G `  y ) }  X.  ( _V 
X.  { y } ) ) " {
x } ) )  =  ( ( _V 
X.  { ( G `
 y ) } )  X.  ( _V 
X.  { y } ) )
3212, 31eqtri 2458 . . . . . 6  |-  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  ( ( _V  X.  { ( G `  y ) } )  X.  ( _V  X.  { y } ) )
3332cnveqi 5029 . . . . 5  |-  `' ( ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  `' ( ( _V 
X.  { ( G `
 y ) } )  X.  ( _V 
X.  { y } ) )
34 cnvco 5040 . . . . 5  |-  `' ( ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  `' ( { ( G `  y ) }  X.  ( _V  X.  { y } ) ) )
35 cnvxp 5274 . . . . 5  |-  `' ( ( _V  X.  {
( G `  y
) } )  X.  ( _V  X.  {
y } ) )  =  ( ( _V 
X.  { y } )  X.  ( _V 
X.  { ( G `
 y ) } ) )
3633, 34, 353eqtr3i 2466 . . . 4  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  `' ( { ( G `  y
) }  X.  ( _V  X.  { y } ) ) )  =  ( ( _V  X.  { y } )  X.  ( _V  X.  { ( G `  y ) } ) )
37 fparlem2 6908 . . . . . . . . 9  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  =  ( _V 
X.  { y } )
3837xpeq2i 4875 . . . . . . . 8  |-  ( { ( G `  y
) }  X.  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } ) )  =  ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )
39 fnsnfv 5941 . . . . . . . . 9  |-  ( ( G  Fn  B  /\  y  e.  B )  ->  { ( G `  y ) }  =  ( G " { y } ) )
4039xpeq1d 4877 . . . . . . . 8  |-  ( ( G  Fn  B  /\  y  e.  B )  ->  ( { ( G `
 y ) }  X.  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " { y } ) )  =  ( ( G " { y } )  X.  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } ) ) )
4138, 40syl5eqr 2484 . . . . . . 7  |-  ( ( G  Fn  B  /\  y  e.  B )  ->  ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  =  ( ( G " { y } )  X.  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } ) ) )
4241cnveqd 5030 . . . . . 6  |-  ( ( G  Fn  B  /\  y  e.  B )  ->  `' ( { ( G `  y ) }  X.  ( _V 
X.  { y } ) )  =  `' ( ( G " { y } )  X.  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " { y } ) ) )
43 cnvxp 5274 . . . . . 6  |-  `' ( ( G " {
y } )  X.  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " {
y } ) )  =  ( ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  X.  ( G
" { y } ) )
4442, 43syl6eq 2486 . . . . 5  |-  ( ( G  Fn  B  /\  y  e.  B )  ->  `' ( { ( G `  y ) }  X.  ( _V 
X.  { y } ) )  =  ( ( `' ( 2nd  |`  ( _V  X.  _V ) ) " {
y } )  X.  ( G " {
y } ) ) )
4544coeq2d 5017 . . . 4  |-  ( ( G  Fn  B  /\  y  e.  B )  ->  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  `' ( { ( G `  y ) }  X.  ( _V  X.  { y } ) ) )  =  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( ( `' ( 2nd  |`  ( _V  X.  _V ) ) " { y } )  X.  ( G " { y } ) ) ) )
4636, 45syl5eqr 2484 . . 3  |-  ( ( G  Fn  B  /\  y  e.  B )  ->  ( ( _V  X.  { y } )  X.  ( _V  X.  { ( G `  y ) } ) )  =  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  X.  ( G
" { y } ) ) ) )
4746iuneq2dv 4324 . 2  |-  ( G  Fn  B  ->  U_ y  e.  B  ( ( _V  X.  { y } )  X.  ( _V 
X.  { ( G `
 y ) } ) )  =  U_ y  e.  B  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  X.  ( G
" { y } ) ) ) )
481, 7, 473eqtr4a 2496 1  |-  ( G  Fn  B  ->  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) ) )  =  U_ y  e.  B  ( ( _V  X.  { y } )  X.  ( _V 
X.  { ( G `
 y ) } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   _Vcvv 3087    i^i cin 3441    C_ wss 3442   (/)c0 3767   {csn 4002   U_ciun 4302    X. cxp 4852   `'ccnv 4853   dom cdm 4854   ran crn 4855    |` cres 4856   "cima 4857    o. ccom 4858    Fn wfn 5596   ` cfv 5601   2ndc2nd 6806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-1st 6807  df-2nd 6808
This theorem is referenced by:  fpar  6911
  Copyright terms: Public domain W3C validator