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Theorem fparlem3 6695
Description: Lemma for fpar 6697. (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 5368 . 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 3591 . . . . 5  |-  ( dom 
F  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  dom  F
3 fndm 5531 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
42, 3syl5sseq 3425 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  A )
5 dfco2a 5359 . . . 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 5023 . 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 3591 . . . . . . . . 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 5290 . . . . . . . . 9  |-  dom  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  C_  { ( F `  x ) }
108, 9sstri 3386 . . . . . . . 8  |-  ( dom  ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  { ( F `  x
) }
11 dfco2a 5359 . . . . . . . 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 5722 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
14 fparlem1 6693 . . . . . . . . . 10  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { y } )  =  ( { y }  X.  _V )
15 sneq 3908 . . . . . . . . . . 11  |-  ( y  =  ( F `  x )  ->  { y }  =  { ( F `  x ) } )
1615xpeq1d 4884 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  ( { y }  X.  _V )  =  ( { ( F `  x ) }  X.  _V ) )
1714, 16syl5eq 2487 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { y } )  =  ( { ( F `  x
) }  X.  _V ) )
1815imaeq2d 5190 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )
" { y } )  =  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } ) )
19 df-ima 4874 . . . . . . . . . . 11  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } )  =  ran  ( ( { ( F `  x
) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )
20 ssid 3396 . . . . . . . . . . . . . 14  |-  { ( F `  x ) }  C_  { ( F `  x ) }
21 xpssres 5165 . . . . . . . . . . . . . 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 5087 . . . . . . . . . . . 12  |-  ran  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ran  ( { ( F `  x
) }  X.  ( { x }  X.  _V ) )
2413snnz 4014 . . . . . . . . . . . . 13  |-  { ( F `  x ) }  =/=  (/)
25 rnxp 5289 . . . . . . . . . . . . 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 2463 . . . . . . . . . . 11  |-  ran  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ( { x }  X.  _V )
2819, 27eqtri 2463 . . . . . . . . . 10  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } )  =  ( { x }  X.  _V )
2918, 28syl6eq 2491 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )
" { y } )  =  ( { x }  X.  _V ) )
3017, 29xpeq12d 4886 . . . . . . . 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 4271 . . . . . . 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 2463 . . . . . 6  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  ( ( { ( F `  x ) }  X.  _V )  X.  ( { x }  X.  _V ) )
3332cnveqi 5035 . . . . 5  |-  `' ( ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  `' ( ( { ( F `
 x ) }  X.  _V )  X.  ( { x }  X.  _V ) )
34 cnvco 5046 . . . . 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 5276 . . . . 5  |-  `' ( ( { ( F `
 x ) }  X.  _V )  X.  ( { x }  X.  _V ) )  =  ( ( { x }  X.  _V )  X.  ( { ( F `
 x ) }  X.  _V ) )
3633, 34, 353eqtr3i 2471 . . . 4  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  `' ( { ( F `  x
) }  X.  ( { x }  X.  _V ) ) )  =  ( ( { x }  X.  _V )  X.  ( { ( F `
 x ) }  X.  _V ) )
37 fparlem1 6693 . . . . . . . . 9  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  =  ( { x }  X.  _V )
3837xpeq2i 4882 . . . . . . . 8  |-  ( { ( F `  x
) }  X.  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
) )  =  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )
39 fnsnfv 5772 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  { ( F `  x ) }  =  ( F " { x } ) )
4039xpeq1d 4884 . . . . . . . 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 2489 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  =  ( ( F
" { x }
)  X.  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
) ) )
4241cnveqd 5036 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  `' ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  `' ( ( F " { x } )  X.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } ) ) )
43 cnvxp 5276 . . . . . 6  |-  `' ( ( F " {
x } )  X.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " {
x } ) )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  X.  ( F
" { x }
) )
4442, 43syl6eq 2491 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  `' ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " {
x } )  X.  ( F " {
x } ) ) )
4544coeq2d 5023 . . . 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 2489 . . 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 4213 . 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 2501 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 1369    e. wcel 1756    =/= wne 2620   _Vcvv 2993    i^i cin 3348    C_ wss 3349   (/)c0 3658   {csn 3898   U_ciun 4192    X. cxp 4859   `'ccnv 4860   dom cdm 4861   ran crn 4862    |` cres 4863   "cima 4864    o. ccom 4865    Fn wfn 5434   ` cfv 5439   1stc1st 6596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-fv 5447  df-1st 6598  df-2nd 6599
This theorem is referenced by:  fpar  6697
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