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Theorem fparlem3 6415
Description: Lemma for fpar 6417. (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 5346 . 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 3529 . . . . 5  |-  ( dom 
F  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  dom  F
3 fndm 5511 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
42, 3syl5sseq 3364 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  A )
5 dfco2a 5337 . . . 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 5002 . 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 3529 . . . . . . . . 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 5267 . . . . . . . . 9  |-  dom  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  C_  { ( F `  x ) }
108, 9sstri 3325 . . . . . . . 8  |-  ( dom  ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  { ( F `  x
) }
11 dfco2a 5337 . . . . . . . 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 5709 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
14 fparlem1 6413 . . . . . . . . . 10  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { y } )  =  ( { y }  X.  _V )
15 sneq 3793 . . . . . . . . . . 11  |-  ( y  =  ( F `  x )  ->  { y }  =  { ( F `  x ) } )
1615xpeq1d 4868 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  ( { y }  X.  _V )  =  ( { ( F `  x ) }  X.  _V ) )
1714, 16syl5eq 2456 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { y } )  =  ( { ( F `  x
) }  X.  _V ) )
1815imaeq2d 5170 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )
" { y } )  =  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } ) )
19 df-ima 4858 . . . . . . . . . . 11  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } )  =  ran  ( ( { ( F `  x
) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )
20 ssid 3335 . . . . . . . . . . . . . 14  |-  { ( F `  x ) }  C_  { ( F `  x ) }
21 xpssres 5147 . . . . . . . . . . . . . 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 5063 . . . . . . . . . . . 12  |-  ran  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ran  ( { ( F `  x
) }  X.  ( { x }  X.  _V ) )
2413snnz 3890 . . . . . . . . . . . . 13  |-  { ( F `  x ) }  =/=  (/)
25 rnxp 5266 . . . . . . . . . . . . 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 2432 . . . . . . . . . . 11  |-  ran  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ( { x }  X.  _V )
2819, 27eqtri 2432 . . . . . . . . . 10  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } )  =  ( { x }  X.  _V )
2918, 28syl6eq 2460 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )
" { y } )  =  ( { x }  X.  _V ) )
3017, 29xpeq12d 4870 . . . . . . . 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 4138 . . . . . . 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 2432 . . . . . 6  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  ( ( { ( F `  x ) }  X.  _V )  X.  ( { x }  X.  _V ) )
3332cnveqi 5014 . . . . 5  |-  `' ( ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  `' ( ( { ( F `
 x ) }  X.  _V )  X.  ( { x }  X.  _V ) )
34 cnvco 5023 . . . . 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 5257 . . . . 5  |-  `' ( ( { ( F `
 x ) }  X.  _V )  X.  ( { x }  X.  _V ) )  =  ( ( { x }  X.  _V )  X.  ( { ( F `
 x ) }  X.  _V ) )
3633, 34, 353eqtr3i 2440 . . . 4  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  `' ( { ( F `  x
) }  X.  ( { x }  X.  _V ) ) )  =  ( ( { x }  X.  _V )  X.  ( { ( F `
 x ) }  X.  _V ) )
37 fparlem1 6413 . . . . . . . . 9  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  =  ( { x }  X.  _V )
3837xpeq2i 4866 . . . . . . . 8  |-  ( { ( F `  x
) }  X.  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
) )  =  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )
39 fnsnfv 5753 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  { ( F `  x ) }  =  ( F " { x } ) )
4039xpeq1d 4868 . . . . . . . 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 2458 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  =  ( ( F
" { x }
)  X.  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
) ) )
4241cnveqd 5015 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  `' ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  `' ( ( F " { x } )  X.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } ) ) )
43 cnvxp 5257 . . . . . 6  |-  `' ( ( F " {
x } )  X.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " {
x } ) )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  X.  ( F
" { x }
) )
4442, 43syl6eq 2460 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  `' ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " {
x } )  X.  ( F " {
x } ) ) )
4544coeq2d 5002 . . . 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 2458 . . 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 4082 . 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 2470 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 359    = wceq 1649    e. wcel 1721    =/= wne 2575   _Vcvv 2924    i^i cin 3287    C_ wss 3288   (/)c0 3596   {csn 3782   U_ciun 4061    X. cxp 4843   `'ccnv 4844   dom cdm 4845   ran crn 4846    |` cres 4847   "cima 4848    o. ccom 4849    Fn wfn 5416   ` cfv 5421   1stc1st 6314
This theorem is referenced by:  fpar  6417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-1st 6316  df-2nd 6317
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