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Theorem fparlem3 6898
Description: Lemma for fpar 6900. (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 5345 . 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 3652 . . . . 5  |-  ( dom 
F  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  dom  F
3 fndm 5675 . . . . 5  |-  ( F  Fn  A  ->  dom  F  =  A )
42, 3syl5sseq 3480 . . . 4  |-  ( F  Fn  A  ->  ( dom  F  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  A )
5 dfco2a 5335 . . . 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 17 . . 3  |-  ( F  Fn  A  ->  ( F  o.  ( 1st  |`  ( _V  X.  _V ) ) )  = 
U_ x  e.  A  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } )  X.  ( F " { x } ) ) )
76coeq2d 4997 . 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 3652 . . . . . . . . 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 5268 . . . . . . . . 9  |-  dom  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  C_  { ( F `  x ) }
108, 9sstri 3441 . . . . . . . 8  |-  ( dom  ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  i^i  ran  ( 1st  |`  ( _V  X.  _V ) ) )  C_  { ( F `  x
) }
11 dfco2a 5335 . . . . . . . 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 5875 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
14 fparlem1 6896 . . . . . . . . . 10  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { y } )  =  ( { y }  X.  _V )
15 sneq 3978 . . . . . . . . . . 11  |-  ( y  =  ( F `  x )  ->  { y }  =  { ( F `  x ) } )
1615xpeq1d 4857 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  ( { y }  X.  _V )  =  ( { ( F `  x ) }  X.  _V ) )
1714, 16syl5eq 2497 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { y } )  =  ( { ( F `  x
) }  X.  _V ) )
1815imaeq2d 5168 . . . . . . . . . 10  |-  ( y  =  ( F `  x )  ->  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )
" { y } )  =  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } ) )
19 df-ima 4847 . . . . . . . . . . 11  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } )  =  ran  ( ( { ( F `  x
) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )
20 ssid 3451 . . . . . . . . . . . . . 14  |-  { ( F `  x ) }  C_  { ( F `  x ) }
21 xpssres 5139 . . . . . . . . . . . . . 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 5061 . . . . . . . . . . . 12  |-  ran  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ran  ( { ( F `  x
) }  X.  ( { x }  X.  _V ) )
2413snnz 4090 . . . . . . . . . . . . 13  |-  { ( F `  x ) }  =/=  (/)
25 rnxp 5267 . . . . . . . . . . . . 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 2473 . . . . . . . . . . 11  |-  ran  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  |`  { ( F `  x ) } )  =  ( { x }  X.  _V )
2819, 27eqtri 2473 . . . . . . . . . 10  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) ) " {
( F `  x
) } )  =  ( { x }  X.  _V )
2918, 28syl6eq 2501 . . . . . . . . 9  |-  ( y  =  ( F `  x )  ->  (
( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )
" { y } )  =  ( { x }  X.  _V ) )
3017, 29xpeq12d 4859 . . . . . . . 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 4361 . . . . . . 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 2473 . . . . . 6  |-  ( ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  ( ( { ( F `  x ) }  X.  _V )  X.  ( { x }  X.  _V ) )
3332cnveqi 5009 . . . . 5  |-  `' ( ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  o.  ( 1st  |`  ( _V  X.  _V ) ) )  =  `' ( ( { ( F `
 x ) }  X.  _V )  X.  ( { x }  X.  _V ) )
34 cnvco 5020 . . . . 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 5254 . . . . 5  |-  `' ( ( { ( F `
 x ) }  X.  _V )  X.  ( { x }  X.  _V ) )  =  ( ( { x }  X.  _V )  X.  ( { ( F `
 x ) }  X.  _V ) )
3633, 34, 353eqtr3i 2481 . . . 4  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )  o.  `' ( { ( F `  x
) }  X.  ( { x }  X.  _V ) ) )  =  ( ( { x }  X.  _V )  X.  ( { ( F `
 x ) }  X.  _V ) )
37 fparlem1 6896 . . . . . . . . 9  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  =  ( { x }  X.  _V )
3837xpeq2i 4855 . . . . . . . 8  |-  ( { ( F `  x
) }  X.  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
) )  =  ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )
39 fnsnfv 5925 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  { ( F `  x ) }  =  ( F " { x } ) )
4039xpeq1d 4857 . . . . . . . 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 2499 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( { ( F `
 x ) }  X.  ( { x }  X.  _V ) )  =  ( ( F
" { x }
)  X.  ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
) ) )
4241cnveqd 5010 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  `' ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  `' ( ( F " { x } )  X.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " { x } ) ) )
43 cnvxp 5254 . . . . . 6  |-  `' ( ( F " {
x } )  X.  ( `' ( 1st  |`  ( _V  X.  _V ) ) " {
x } ) )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) )
" { x }
)  X.  ( F
" { x }
) )
4442, 43syl6eq 2501 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  `' ( { ( F `  x ) }  X.  ( { x }  X.  _V ) )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " {
x } )  X.  ( F " {
x } ) ) )
4544coeq2d 4997 . . . 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 2499 . . 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 4300 . 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 2511 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 371    = wceq 1444    e. wcel 1887    =/= wne 2622   _Vcvv 3045    i^i cin 3403    C_ wss 3404   (/)c0 3731   {csn 3968   U_ciun 4278    X. cxp 4832   `'ccnv 4833   dom cdm 4834   ran crn 4835    |` cres 4836   "cima 4837    o. ccom 4838    Fn wfn 5577   ` cfv 5582   1stc1st 6791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-1st 6793  df-2nd 6794
This theorem is referenced by:  fpar  6900
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