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Theorem fparlem4 6918
Description: Lemma for fpar 6919. (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 5352 . 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 3643 . . . . 5  |-  ( dom 
G  i^i  ran  ( 2nd  |`  ( _V  X.  _V ) ) )  C_  dom  G
3 fndm 5685 . . . . 5  |-  ( G  Fn  B  ->  dom  G  =  B )
42, 3syl5sseq 3466 . . . 4  |-  ( G  Fn  B  ->  ( dom  G  i^i  ran  ( 2nd  |`  ( _V  X.  _V ) ) )  C_  B )
5 dfco2a 5342 . . . 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 5002 . 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 3643 . . . . . . . . 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 5274 . . . . . . . . 9  |-  dom  ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  C_  { ( G `  y
) }
108, 9sstri 3427 . . . . . . . 8  |-  ( dom  ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  i^i  ran  ( 2nd  |`  ( _V  X.  _V ) ) )  C_  { ( G `  y
) }
11 dfco2a 5342 . . . . . . . 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 5889 . . . . . . . 8  |-  ( G `
 y )  e. 
_V
14 fparlem2 6916 . . . . . . . . . 10  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { x }
)  =  ( _V 
X.  { x }
)
15 sneq 3969 . . . . . . . . . . 11  |-  ( x  =  ( G `  y )  ->  { x }  =  { ( G `  y ) } )
1615xpeq2d 4863 . . . . . . . . . 10  |-  ( x  =  ( G `  y )  ->  ( _V  X.  { x }
)  =  ( _V 
X.  { ( G `
 y ) } ) )
1714, 16syl5eq 2517 . . . . . . . . 9  |-  ( x  =  ( G `  y )  ->  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { x }
)  =  ( _V 
X.  { ( G `
 y ) } ) )
1815imaeq2d 5174 . . . . . . . . . 10  |-  ( x  =  ( G `  y )  ->  (
( { ( G `
 y ) }  X.  ( _V  X.  { y } ) ) " { x } )  =  ( ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) ) " { ( G `  y ) } ) )
19 df-ima 4852 . . . . . . . . . . 11  |-  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) ) " { ( G `  y ) } )  =  ran  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  |`  { ( G `  y ) } )
20 ssid 3437 . . . . . . . . . . . . . 14  |-  { ( G `  y ) }  C_  { ( G `  y ) }
21 xpssres 5145 . . . . . . . . . . . . . 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 5067 . . . . . . . . . . . 12  |-  ran  (
( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  |`  { ( G `  y ) } )  =  ran  ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )
2413snnz 4081 . . . . . . . . . . . . 13  |-  { ( G `  y ) }  =/=  (/)
25 rnxp 5273 . . . . . . . . . . . . 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 2493 . . . . . . . . . . 11  |-  ran  (
( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  |`  { ( G `  y ) } )  =  ( _V  X.  { y } )
2819, 27eqtri 2493 . . . . . . . . . 10  |-  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) ) " { ( G `  y ) } )  =  ( _V  X.  { y } )
2918, 28syl6eq 2521 . . . . . . . . 9  |-  ( x  =  ( G `  y )  ->  (
( { ( G `
 y ) }  X.  ( _V  X.  { y } ) ) " { x } )  =  ( _V  X.  { y } ) )
3017, 29xpeq12d 4864 . . . . . . . 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 4352 . . . . . . 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 2493 . . . . . 6  |-  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  ( ( _V  X.  { ( G `  y ) } )  X.  ( _V  X.  { y } ) )
3332cnveqi 5014 . . . . 5  |-  `' ( ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  `' ( ( _V 
X.  { ( G `
 y ) } )  X.  ( _V 
X.  { y } ) )
34 cnvco 5025 . . . . 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 5260 . . . . 5  |-  `' ( ( _V  X.  {
( G `  y
) } )  X.  ( _V  X.  {
y } ) )  =  ( ( _V 
X.  { y } )  X.  ( _V 
X.  { ( G `
 y ) } ) )
3633, 34, 353eqtr3i 2501 . . . 4  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )  o.  `' ( { ( G `  y
) }  X.  ( _V  X.  { y } ) ) )  =  ( ( _V  X.  { y } )  X.  ( _V  X.  { ( G `  y ) } ) )
37 fparlem2 6916 . . . . . . . . 9  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  =  ( _V 
X.  { y } )
3837xpeq2i 4860 . . . . . . . 8  |-  ( { ( G `  y
) }  X.  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } ) )  =  ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )
39 fnsnfv 5940 . . . . . . . . 9  |-  ( ( G  Fn  B  /\  y  e.  B )  ->  { ( G `  y ) }  =  ( G " { y } ) )
4039xpeq1d 4862 . . . . . . . 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 2519 . . . . . . 7  |-  ( ( G  Fn  B  /\  y  e.  B )  ->  ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  =  ( ( G " { y } )  X.  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } ) ) )
4241cnveqd 5015 . . . . . 6  |-  ( ( G  Fn  B  /\  y  e.  B )  ->  `' ( { ( G `  y ) }  X.  ( _V 
X.  { y } ) )  =  `' ( ( G " { y } )  X.  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " { y } ) ) )
43 cnvxp 5260 . . . . . 6  |-  `' ( ( G " {
y } )  X.  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " {
y } ) )  =  ( ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  X.  ( G
" { y } ) )
4442, 43syl6eq 2521 . . . . 5  |-  ( ( G  Fn  B  /\  y  e.  B )  ->  `' ( { ( G `  y ) }  X.  ( _V 
X.  { y } ) )  =  ( ( `' ( 2nd  |`  ( _V  X.  _V ) ) " {
y } )  X.  ( G " {
y } ) ) )
4544coeq2d 5002 . . . 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 2519 . . 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 4291 . 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 2531 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 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    i^i cin 3389    C_ wss 3390   (/)c0 3722   {csn 3959   U_ciun 4269    X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840    |` cres 4841   "cima 4842    o. ccom 4843    Fn wfn 5584   ` cfv 5589   2ndc2nd 6811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-1st 6812  df-2nd 6813
This theorem is referenced by:  fpar  6919
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