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Theorem fparlem3 6220
Description: Lemma for fpar 6222. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fparlem3  |-  ( F  Fn  A  ->  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  =  U_ x  e.  A  ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) ) )
Distinct variable groups:    x, A    x, F

Proof of Theorem fparlem3
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 coiun 5182 . 2  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  U_ x  e.  A  ( ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  X.  ( F
" { x }
) ) )  = 
U_ x  e.  A  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  X.  ( F
" { x }
) ) )
2 inss1 3389 . . . . 5  |-  ( dom 
F  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  dom  F
3 fndm 5343 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
42, 3syl5sseq 3226 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  A )
5 dfco2a 5173 . . . 4  |-  ( ( dom  F  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) ) 
C_  A  ->  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) )  = 
U_ x  e.  A  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } )  X.  ( F " { x } ) ) )
64, 5syl 15 . . 3  |-  ( F  Fn  A  ->  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) )  = 
U_ x  e.  A  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } )  X.  ( F " { x } ) ) )
76coeq2d 4846 . 2  |-  ( F  Fn  A  ->  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  =  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  U_ x  e.  A  ( ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  X.  ( F
" { x }
) ) ) )
8 inss1 3389 . . . . . . . . 9  |-  ( dom  ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  dom  ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )
9 dmxpss 5107 . . . . . . . . 9  |-  dom  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  C_  { ( F `  x ) }
108, 9sstri 3188 . . . . . . . 8  |-  ( dom  ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  { ( F `  x
) }
11 dfco2a 5173 . . . . . . . 8  |-  ( ( dom  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) ) 
C_  { ( F `
 x ) }  ->  ( ( { ( F `  x
) }  X.  ( { x }  X.  _V ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  = 
U_ y  e.  {
( F `  x
) }  ( ( `' ( 1st  |`  ( _V  X.  _V ) )
" { y } )  X.  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
y } ) ) )
1210, 11ax-mp 8 . . . . . . 7  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  = 
U_ y  e.  {
( F `  x
) }  ( ( `' ( 1st  |`  ( _V  X.  _V ) )
" { y } )  X.  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
y } ) )
13 fvex 5539 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
14 fparlem1 6218 . . . . . . . . . 10  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { y } )  =  ( { y }  X.  _V )
15 sneq 3651 . . . . . . . . . . 11  |-  ( y  =  ( F `  x )  ->  { y }  =  { ( F `  x ) } )
1615xpeq1d 4712 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  ( { y }  X.  _V )  =  ( { ( F `  x ) }  X.  _V ) )
1714, 16syl5eq 2327 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { y } )  =  ( { ( F `  x
) }  X.  _V ) )
1815imaeq2d 5012 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )
" { y } )  =  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } ) )
19 df-ima 4702 . . . . . . . . . . 11  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } )  =  ran  ( ( { ( F `  x
) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )
20 ssid 3197 . . . . . . . . . . . . . 14  |-  { ( F `  x ) }  C_  { ( F `  x ) }
21 xpssres 4989 . . . . . . . . . . . . . 14  |-  ( { ( F `  x
) }  C_  { ( F `  x ) }  ->  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) )
2220, 21ax-mp 8 . . . . . . . . . . . . 13  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )
2322rneqi 4905 . . . . . . . . . . . 12  |-  ran  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ran  ( { ( F `  x
) }  X.  ( { x }  X.  _V ) )
2413snnz 3744 . . . . . . . . . . . . 13  |-  { ( F `  x ) }  =/=  (/)
25 rnxp 5106 . . . . . . . . . . . . 13  |-  ( { ( F `  x
) }  =/=  (/)  ->  ran  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  ( { x }  X.  _V ) )
2624, 25ax-mp 8 . . . . . . . . . . . 12  |-  ran  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  ( { x }  X.  _V )
2723, 26eqtri 2303 . . . . . . . . . . 11  |-  ran  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ( { x }  X.  _V )
2819, 27eqtri 2303 . . . . . . . . . 10  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } )  =  ( { x }  X.  _V )
2918, 28syl6eq 2331 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )
" { y } )  =  ( { x }  X.  _V ) )
3017, 29xpeq12d 4714 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
( `' ( 1st  |`  ( _V  X.  _V ) ) " {
y } )  X.  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
y } ) )  =  ( ( { ( F `  x
) }  X.  _V )  X.  ( { x }  X.  _V ) ) )
3113, 30iunxsn 3981 . . . . . . 7  |-  U_ y  e.  { ( F `  x ) }  (
( `' ( 1st  |`  ( _V  X.  _V ) ) " {
y } )  X.  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
y } ) )  =  ( ( { ( F `  x
) }  X.  _V )  X.  ( { x }  X.  _V ) )
3212, 31eqtri 2303 . . . . . 6  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  ( ( { ( F `  x ) }  X.  _V )  X.  ( { x }  X.  _V ) )
3332cnveqi 4856 . . . . 5  |-  `' ( ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  `' ( ( { ( F `
 x ) }  X.  _V )  X.  ( { x }  X.  _V ) )
34 cnvco 4865 . . . . 5  |-  `' ( ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  `' ( { ( F `  x
) }  X.  ( { x }  X.  _V ) ) )
35 cnvxp 5097 . . . . 5  |-  `' ( ( { ( F `
 x ) }  X.  _V )  X.  ( { x }  X.  _V ) )  =  ( ( { x }  X.  _V )  X.  ( { ( F `
 x ) }  X.  _V ) )
3633, 34, 353eqtr3i 2311 . . . 4  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  `' ( { ( F `  x
) }  X.  ( { x }  X.  _V ) ) )  =  ( ( { x }  X.  _V )  X.  ( { ( F `
 x ) }  X.  _V ) )
37 fparlem1 6218 . . . . . . . . 9  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  =  ( { x }  X.  _V )
3837xpeq2i 4710 . . . . . . . 8  |-  ( { ( F `  x
) }  X.  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
) )  =  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )
39 fnsnfv 5582 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  { ( F `  x ) }  =  ( F " { x } ) )
4039xpeq1d 4712 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( { ( F `
 x ) }  X.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } ) )  =  ( ( F " { x } )  X.  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
) ) )
4138, 40syl5eqr 2329 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  =  ( ( F
" { x }
)  X.  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
) ) )
4241cnveqd 4857 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  `' ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  `' ( ( F " { x } )  X.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } ) ) )
43 cnvxp 5097 . . . . . 6  |-  `' ( ( F " {
x } )  X.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " {
x } ) )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  X.  ( F
" { x }
) )
4442, 43syl6eq 2331 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  `' ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " {
x } )  X.  ( F " {
x } ) ) )
4544coeq2d 4846 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  `' ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) )  =  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  (
( `' ( 1st  |`  ( _V  X.  _V ) ) " {
x } )  X.  ( F " {
x } ) ) ) )
4636, 45syl5eqr 2329 . . 3  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( { x }  X.  _V )  X.  ( { ( F `
 x ) }  X.  _V ) )  =  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } )  X.  ( F " { x } ) ) ) )
4746iuneq2dv 3926 . 2  |-  ( F  Fn  A  ->  U_ x  e.  A  ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) )  =  U_ x  e.  A  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  X.  ( F
" { x }
) ) ) )
481, 7, 473eqtr4a 2341 1  |-  ( F  Fn  A  ->  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) ) )  =  U_ x  e.  A  ( ( { x }  X.  _V )  X.  ( { ( F `  x ) }  X.  _V ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   U_ciun 3905    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693    Fn wfn 5250   ` cfv 5255   1stc1st 6120
This theorem is referenced by:  fpar  6222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-1st 6122  df-2nd 6123
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