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Theorem fparlem3 6901
Description: Lemma for fpar 6903. (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 5523 . 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 3714 . . . . 5  |-  ( dom 
F  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  dom  F
3 fndm 5686 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
42, 3syl5sseq 3547 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  A )
5 dfco2a 5513 . . . 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 16 . . 3  |-  ( F  Fn  A  ->  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) )  = 
U_ x  e.  A  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } )  X.  ( F " { x } ) ) )
76coeq2d 5175 . 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 3714 . . . . . . . . 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 5445 . . . . . . . . 9  |-  dom  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  C_  { ( F `  x ) }
108, 9sstri 3508 . . . . . . . 8  |-  ( dom  ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  { ( F `  x
) }
11 dfco2a 5513 . . . . . . . 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 5 . . . . . . 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 5882 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
14 fparlem1 6899 . . . . . . . . . 10  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { y } )  =  ( { y }  X.  _V )
15 sneq 4042 . . . . . . . . . . 11  |-  ( y  =  ( F `  x )  ->  { y }  =  { ( F `  x ) } )
1615xpeq1d 5031 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  ( { y }  X.  _V )  =  ( { ( F `  x ) }  X.  _V ) )
1714, 16syl5eq 2510 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { y } )  =  ( { ( F `  x
) }  X.  _V ) )
1815imaeq2d 5347 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )
" { y } )  =  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } ) )
19 df-ima 5021 . . . . . . . . . . 11  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } )  =  ran  ( ( { ( F `  x
) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )
20 ssid 3518 . . . . . . . . . . . . . 14  |-  { ( F `  x ) }  C_  { ( F `  x ) }
21 xpssres 5318 . . . . . . . . . . . . . 14  |-  ( { ( F `  x
) }  C_  { ( F `  x ) }  ->  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) )
2220, 21ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )
2322rneqi 5239 . . . . . . . . . . . 12  |-  ran  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ran  ( { ( F `  x
) }  X.  ( { x }  X.  _V ) )
2413snnz 4150 . . . . . . . . . . . . 13  |-  { ( F `  x ) }  =/=  (/)
25 rnxp 5444 . . . . . . . . . . . . 13  |-  ( { ( F `  x
) }  =/=  (/)  ->  ran  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  ( { x }  X.  _V ) )
2624, 25ax-mp 5 . . . . . . . . . . . 12  |-  ran  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  ( { x }  X.  _V )
2723, 26eqtri 2486 . . . . . . . . . . 11  |-  ran  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ( { x }  X.  _V )
2819, 27eqtri 2486 . . . . . . . . . 10  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } )  =  ( { x }  X.  _V )
2918, 28syl6eq 2514 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )
" { y } )  =  ( { x }  X.  _V ) )
3017, 29xpeq12d 5033 . . . . . . . 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 4415 . . . . . . 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 2486 . . . . . 6  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  ( ( { ( F `  x ) }  X.  _V )  X.  ( { x }  X.  _V ) )
3332cnveqi 5187 . . . . 5  |-  `' ( ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  `' ( ( { ( F `
 x ) }  X.  _V )  X.  ( { x }  X.  _V ) )
34 cnvco 5198 . . . . 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 5431 . . . . 5  |-  `' ( ( { ( F `
 x ) }  X.  _V )  X.  ( { x }  X.  _V ) )  =  ( ( { x }  X.  _V )  X.  ( { ( F `
 x ) }  X.  _V ) )
3633, 34, 353eqtr3i 2494 . . . 4  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  `' ( { ( F `  x
) }  X.  ( { x }  X.  _V ) ) )  =  ( ( { x }  X.  _V )  X.  ( { ( F `
 x ) }  X.  _V ) )
37 fparlem1 6899 . . . . . . . . 9  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  =  ( { x }  X.  _V )
3837xpeq2i 5029 . . . . . . . 8  |-  ( { ( F `  x
) }  X.  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
) )  =  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )
39 fnsnfv 5933 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  { ( F `  x ) }  =  ( F " { x } ) )
4039xpeq1d 5031 . . . . . . . 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 2512 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  =  ( ( F
" { x }
)  X.  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
) ) )
4241cnveqd 5188 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  `' ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  `' ( ( F " { x } )  X.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } ) ) )
43 cnvxp 5431 . . . . . 6  |-  `' ( ( F " {
x } )  X.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " {
x } ) )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  X.  ( F
" { x }
) )
4442, 43syl6eq 2514 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  `' ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " {
x } )  X.  ( F " {
x } ) ) )
4544coeq2d 5175 . . . 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 2512 . . 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 4354 . 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 2524 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109    i^i cin 3470    C_ wss 3471   (/)c0 3793   {csn 4032   U_ciun 4332    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011    o. ccom 5012    Fn wfn 5589   ` cfv 5594   1stc1st 6797
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-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-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-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  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-fv 5602  df-1st 6799  df-2nd 6800
This theorem is referenced by:  fpar  6903
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