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Theorem fparlem4 6451
Description: Lemma for fpar 6452. (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 5381 . 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 3563 . . . . 5  |-  ( dom 
G  i^i  ran  ( 2nd  |`  ( _V  X.  _V ) ) )  C_  dom  G
3 fndm 5546 . . . . 5  |-  ( G  Fn  B  ->  dom  G  =  B )
42, 3syl5sseq 3398 . . . 4  |-  ( G  Fn  B  ->  ( dom  G  i^i  ran  ( 2nd  |`  ( _V  X.  _V ) ) )  C_  B )
5 dfco2a 5372 . . . 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 16 . . 3  |-  ( G  Fn  B  ->  ( G  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  = 
U_ y  e.  B  ( ( `' ( 2nd  |`  ( _V  X.  _V ) ) " { y } )  X.  ( G " { y } ) ) )
76coeq2d 5037 . 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 3563 . . . . . . . . 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 5302 . . . . . . . . 9  |-  dom  ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  C_  { ( G `  y
) }
108, 9sstri 3359 . . . . . . . 8  |-  ( dom  ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  i^i  ran  ( 2nd  |`  ( _V  X.  _V ) ) )  C_  { ( G `  y
) }
11 dfco2a 5372 . . . . . . . 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 8 . . . . . . 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 5744 . . . . . . . 8  |-  ( G `
 y )  e. 
_V
14 fparlem2 6449 . . . . . . . . . 10  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { x }
)  =  ( _V 
X.  { x }
)
15 sneq 3827 . . . . . . . . . . 11  |-  ( x  =  ( G `  y )  ->  { x }  =  { ( G `  y ) } )
1615xpeq2d 4904 . . . . . . . . . 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 5205 . . . . . . . . . 10  |-  ( x  =  ( G `  y )  ->  (
( { ( G `
 y ) }  X.  ( _V  X.  { y } ) ) " { x } )  =  ( ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) ) " { ( G `  y ) } ) )
19 df-ima 4893 . . . . . . . . . . 11  |-  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) ) " { ( G `  y ) } )  =  ran  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  |`  { ( G `  y ) } )
20 ssid 3369 . . . . . . . . . . . . . 14  |-  { ( G `  y ) }  C_  { ( G `  y ) }
21 xpssres 5182 . . . . . . . . . . . . . 14  |-  ( { ( G `  y
) }  C_  { ( G `  y ) }  ->  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  |`  { ( G `  y ) } )  =  ( { ( G `  y ) }  X.  ( _V 
X.  { y } ) ) )
2220, 21ax-mp 8 . . . . . . . . . . . . 13  |-  ( ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  |`  { ( G `  y ) } )  =  ( { ( G `  y ) }  X.  ( _V 
X.  { y } ) )
2322rneqi 5098 . . . . . . . . . . . 12  |-  ran  (
( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  |`  { ( G `  y ) } )  =  ran  ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )
2413snnz 3924 . . . . . . . . . . . . 13  |-  { ( G `  y ) }  =/=  (/)
25 rnxp 5301 . . . . . . . . . . . . 13  |-  ( { ( G `  y
) }  =/=  (/)  ->  ran  ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )  =  ( _V  X.  {
y } ) )
2624, 25ax-mp 8 . . . . . . . . . . . 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 4905 . . . . . . . 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 4172 . . . . . . 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 5049 . . . . 5  |-  `' ( ( { ( G `
 y ) }  X.  ( _V  X.  { y } ) )  o.  ( 2nd  |`  ( _V  X.  _V ) ) )  =  `' ( ( _V 
X.  { ( G `
 y ) } )  X.  ( _V 
X.  { y } ) )
34 cnvco 5058 . . . . 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 5292 . . . . 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 6449 . . . . . . . . 9  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } )  =  ( _V 
X.  { y } )
3837xpeq2i 4901 . . . . . . . 8  |-  ( { ( G `  y
) }  X.  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" { y } ) )  =  ( { ( G `  y ) }  X.  ( _V  X.  { y } ) )
39 fnsnfv 5788 . . . . . . . . 9  |-  ( ( G  Fn  B  /\  y  e.  B )  ->  { ( G `  y ) }  =  ( G " { y } ) )
4039xpeq1d 4903 . . . . . . . 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 5050 . . . . . 6  |-  ( ( G  Fn  B  /\  y  e.  B )  ->  `' ( { ( G `  y ) }  X.  ( _V 
X.  { y } ) )  =  `' ( ( G " { y } )  X.  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " { y } ) ) )
43 cnvxp 5292 . . . . . 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 5037 . . . 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 4116 . 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958    i^i cin 3321    C_ wss 3322   (/)c0 3630   {csn 3816   U_ciun 4095    X. cxp 4878   `'ccnv 4879   dom cdm 4880   ran crn 4881    |` cres 4882   "cima 4883    o. ccom 4884    Fn wfn 5451   ` cfv 5456   2ndc2nd 6350
This theorem is referenced by:  fpar  6452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-1st 6351  df-2nd 6352
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