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Theorem fmucndlem 21237
Description: Lemma for fmucnd 21238. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Assertion
Ref Expression
fmucndlem  |-  ( ( F  Fn  X  /\  A  C_  X )  -> 
( ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " ( A  X.  A ) )  =  ( ( F
" A )  X.  ( F " A
) ) )
Distinct variable groups:    x, y, A    x, F, y    x, X, y

Proof of Theorem fmucndlem
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 df-ima 4867 . . 3  |-  ( ( x  e.  X , 
y  e.  X  |->  <.
( F `  x
) ,  ( F `
 y ) >.
) " ( A  X.  A ) )  =  ran  ( ( x  e.  X , 
y  e.  X  |->  <.
( F `  x
) ,  ( F `
 y ) >.
)  |`  ( A  X.  A ) )
2 simpr 462 . . . . 5  |-  ( ( F  Fn  X  /\  A  C_  X )  ->  A  C_  X )
3 resmpt2 6408 . . . . 5  |-  ( ( A  C_  X  /\  A  C_  X )  -> 
( ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. )  |`  ( A  X.  A ) )  =  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y )
>. ) )
42, 3sylancom 671 . . . 4  |-  ( ( F  Fn  X  /\  A  C_  X )  -> 
( ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. )  |`  ( A  X.  A ) )  =  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y )
>. ) )
54rneqd 5082 . . 3  |-  ( ( F  Fn  X  /\  A  C_  X )  ->  ran  ( ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. )  |`  ( A  X.  A ) )  =  ran  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y ) >. )
)
61, 5syl5eq 2482 . 2  |-  ( ( F  Fn  X  /\  A  C_  X )  -> 
( ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " ( A  X.  A ) )  =  ran  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y ) >. )
)
7 vex 3090 . . . . . . . . . . . . 13  |-  x  e. 
_V
8 vex 3090 . . . . . . . . . . . . 13  |-  y  e. 
_V
97, 8op1std 6817 . . . . . . . . . . . 12  |-  ( p  =  <. x ,  y
>.  ->  ( 1st `  p
)  =  x )
109fveq2d 5885 . . . . . . . . . . 11  |-  ( p  =  <. x ,  y
>.  ->  ( F `  ( 1st `  p ) )  =  ( F `
 x ) )
117, 8op2ndd 6818 . . . . . . . . . . . 12  |-  ( p  =  <. x ,  y
>.  ->  ( 2nd `  p
)  =  y )
1211fveq2d 5885 . . . . . . . . . . 11  |-  ( p  =  <. x ,  y
>.  ->  ( F `  ( 2nd `  p ) )  =  ( F `
 y ) )
1310, 12opeq12d 4198 . . . . . . . . . 10  |-  ( p  =  <. x ,  y
>.  ->  <. ( F `  ( 1st `  p ) ) ,  ( F `
 ( 2nd `  p
) ) >.  =  <. ( F `  x ) ,  ( F `  y ) >. )
1413mpt2mpt 6402 . . . . . . . . 9  |-  ( p  e.  ( A  X.  A )  |->  <. ( F `  ( 1st `  p ) ) ,  ( F `  ( 2nd `  p ) )
>. )  =  (
x  e.  A , 
y  e.  A  |->  <.
( F `  x
) ,  ( F `
 y ) >.
)
1514eqcomi 2442 . . . . . . . 8  |-  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y ) >. )  =  ( p  e.  ( A  X.  A
)  |->  <. ( F `  ( 1st `  p ) ) ,  ( F `
 ( 2nd `  p
) ) >. )
1615rneqi 5081 . . . . . . 7  |-  ran  (
x  e.  A , 
y  e.  A  |->  <.
( F `  x
) ,  ( F `
 y ) >.
)  =  ran  (
p  e.  ( A  X.  A )  |->  <.
( F `  ( 1st `  p ) ) ,  ( F `  ( 2nd `  p ) ) >. )
17 fvex 5891 . . . . . . . 8  |-  ( F `
 ( 1st `  p
) )  e.  _V
1817a1i 11 . . . . . . 7  |-  ( ( T.  /\  p  e.  ( A  X.  A
) )  ->  ( F `  ( 1st `  p ) )  e. 
_V )
19 fvex 5891 . . . . . . . 8  |-  ( F `
 ( 2nd `  p
) )  e.  _V
2019a1i 11 . . . . . . 7  |-  ( ( T.  /\  p  e.  ( A  X.  A
) )  ->  ( F `  ( 2nd `  p ) )  e. 
_V )
2116, 18, 20fliftrel 6216 . . . . . 6  |-  ( T. 
->  ran  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y )
>. )  C_  ( _V 
X.  _V ) )
2221trud 1446 . . . . 5  |-  ran  (
x  e.  A , 
y  e.  A  |->  <.
( F `  x
) ,  ( F `
 y ) >.
)  C_  ( _V  X.  _V )
2322sseli 3466 . . . 4  |-  ( p  e.  ran  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y ) >. )  ->  p  e.  ( _V 
X.  _V ) )
2423adantl 467 . . 3  |-  ( ( ( F  Fn  X  /\  A  C_  X )  /\  p  e.  ran  ( x  e.  A ,  y  e.  A  |-> 
<. ( F `  x
) ,  ( F `
 y ) >.
) )  ->  p  e.  ( _V  X.  _V ) )
25 xpss 4961 . . . . 5  |-  ( ( F " A )  X.  ( F " A ) )  C_  ( _V  X.  _V )
2625sseli 3466 . . . 4  |-  ( p  e.  ( ( F
" A )  X.  ( F " A
) )  ->  p  e.  ( _V  X.  _V ) )
2726adantl 467 . . 3  |-  ( ( ( F  Fn  X  /\  A  C_  X )  /\  p  e.  ( ( F " A
)  X.  ( F
" A ) ) )  ->  p  e.  ( _V  X.  _V )
)
28 fvelimab 5937 . . . . . . . 8  |-  ( ( F  Fn  X  /\  A  C_  X )  -> 
( ( 1st `  p
)  e.  ( F
" A )  <->  E. x  e.  A  ( F `  x )  =  ( 1st `  p ) ) )
29 fvelimab 5937 . . . . . . . 8  |-  ( ( F  Fn  X  /\  A  C_  X )  -> 
( ( 2nd `  p
)  e.  ( F
" A )  <->  E. y  e.  A  ( F `  y )  =  ( 2nd `  p ) ) )
3028, 29anbi12d 715 . . . . . . 7  |-  ( ( F  Fn  X  /\  A  C_  X )  -> 
( ( ( 1st `  p )  e.  ( F " A )  /\  ( 2nd `  p
)  e.  ( F
" A ) )  <-> 
( E. x  e.  A  ( F `  x )  =  ( 1st `  p )  /\  E. y  e.  A  ( F `  y )  =  ( 2nd `  p ) ) ) )
31 eqid 2429 . . . . . . . . 9  |-  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y ) >. )  =  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y )
>. )
32 opex 4686 . . . . . . . . 9  |-  <. ( F `  x ) ,  ( F `  y ) >.  e.  _V
3331, 32elrnmpt2 6423 . . . . . . . 8  |-  ( <.
( 1st `  p
) ,  ( 2nd `  p ) >.  e.  ran  ( x  e.  A ,  y  e.  A  |-> 
<. ( F `  x
) ,  ( F `
 y ) >.
)  <->  E. x  e.  A  E. y  e.  A  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  =  <. ( F `  x ) ,  ( F `  y ) >. )
34 eqcom 2438 . . . . . . . . . 10  |-  ( <.
( 1st `  p
) ,  ( 2nd `  p ) >.  =  <. ( F `  x ) ,  ( F `  y ) >.  <->  <. ( F `
 x ) ,  ( F `  y
) >.  =  <. ( 1st `  p ) ,  ( 2nd `  p
) >. )
35 fvex 5891 . . . . . . . . . . 11  |-  ( 1st `  p )  e.  _V
36 fvex 5891 . . . . . . . . . . 11  |-  ( 2nd `  p )  e.  _V
3735, 36opth2 4700 . . . . . . . . . 10  |-  ( <.
( F `  x
) ,  ( F `
 y ) >.  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  <->  ( ( F `  x )  =  ( 1st `  p
)  /\  ( F `  y )  =  ( 2nd `  p ) ) )
3834, 37bitri 252 . . . . . . . . 9  |-  ( <.
( 1st `  p
) ,  ( 2nd `  p ) >.  =  <. ( F `  x ) ,  ( F `  y ) >.  <->  ( ( F `  x )  =  ( 1st `  p
)  /\  ( F `  y )  =  ( 2nd `  p ) ) )
39382rexbii 2935 . . . . . . . 8  |-  ( E. x  e.  A  E. y  e.  A  <. ( 1st `  p ) ,  ( 2nd `  p
) >.  =  <. ( F `  x ) ,  ( F `  y ) >.  <->  E. x  e.  A  E. y  e.  A  ( ( F `  x )  =  ( 1st `  p
)  /\  ( F `  y )  =  ( 2nd `  p ) ) )
40 reeanv 3003 . . . . . . . 8  |-  ( E. x  e.  A  E. y  e.  A  (
( F `  x
)  =  ( 1st `  p )  /\  ( F `  y )  =  ( 2nd `  p
) )  <->  ( E. x  e.  A  ( F `  x )  =  ( 1st `  p
)  /\  E. y  e.  A  ( F `  y )  =  ( 2nd `  p ) ) )
4133, 39, 403bitri 274 . . . . . . 7  |-  ( <.
( 1st `  p
) ,  ( 2nd `  p ) >.  e.  ran  ( x  e.  A ,  y  e.  A  |-> 
<. ( F `  x
) ,  ( F `
 y ) >.
)  <->  ( E. x  e.  A  ( F `  x )  =  ( 1st `  p )  /\  E. y  e.  A  ( F `  y )  =  ( 2nd `  p ) ) )
4230, 41syl6rbbr 267 . . . . . 6  |-  ( ( F  Fn  X  /\  A  C_  X )  -> 
( <. ( 1st `  p
) ,  ( 2nd `  p ) >.  e.  ran  ( x  e.  A ,  y  e.  A  |-> 
<. ( F `  x
) ,  ( F `
 y ) >.
)  <->  ( ( 1st `  p )  e.  ( F " A )  /\  ( 2nd `  p
)  e.  ( F
" A ) ) ) )
43 opelxp 4884 . . . . . 6  |-  ( <.
( 1st `  p
) ,  ( 2nd `  p ) >.  e.  ( ( F " A
)  X.  ( F
" A ) )  <-> 
( ( 1st `  p
)  e.  ( F
" A )  /\  ( 2nd `  p )  e.  ( F " A ) ) )
4442, 43syl6bbr 266 . . . . 5  |-  ( ( F  Fn  X  /\  A  C_  X )  -> 
( <. ( 1st `  p
) ,  ( 2nd `  p ) >.  e.  ran  ( x  e.  A ,  y  e.  A  |-> 
<. ( F `  x
) ,  ( F `
 y ) >.
)  <->  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  e.  ( ( F " A
)  X.  ( F
" A ) ) ) )
4544adantr 466 . . . 4  |-  ( ( ( F  Fn  X  /\  A  C_  X )  /\  p  e.  ( _V  X.  _V )
)  ->  ( <. ( 1st `  p ) ,  ( 2nd `  p
) >.  e.  ran  (
x  e.  A , 
y  e.  A  |->  <.
( F `  x
) ,  ( F `
 y ) >.
)  <->  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  e.  ( ( F " A
)  X.  ( F
" A ) ) ) )
46 1st2nd2 6844 . . . . . 6  |-  ( p  e.  ( _V  X.  _V )  ->  p  = 
<. ( 1st `  p
) ,  ( 2nd `  p ) >. )
4746adantl 467 . . . . 5  |-  ( ( ( F  Fn  X  /\  A  C_  X )  /\  p  e.  ( _V  X.  _V )
)  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
4847eleq1d 2498 . . . 4  |-  ( ( ( F  Fn  X  /\  A  C_  X )  /\  p  e.  ( _V  X.  _V )
)  ->  ( p  e.  ran  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y )
>. )  <->  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  e.  ran  ( x  e.  A ,  y  e.  A  |-> 
<. ( F `  x
) ,  ( F `
 y ) >.
) ) )
4947eleq1d 2498 . . . 4  |-  ( ( ( F  Fn  X  /\  A  C_  X )  /\  p  e.  ( _V  X.  _V )
)  ->  ( p  e.  ( ( F " A )  X.  ( F " A ) )  <->  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  e.  ( ( F " A
)  X.  ( F
" A ) ) ) )
5045, 48, 493bitr4d 288 . . 3  |-  ( ( ( F  Fn  X  /\  A  C_  X )  /\  p  e.  ( _V  X.  _V )
)  ->  ( p  e.  ran  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y )
>. )  <->  p  e.  (
( F " A
)  X.  ( F
" A ) ) ) )
5124, 27, 50eqrdav 2427 . 2  |-  ( ( F  Fn  X  /\  A  C_  X )  ->  ran  ( x  e.  A ,  y  e.  A  |-> 
<. ( F `  x
) ,  ( F `
 y ) >.
)  =  ( ( F " A )  X.  ( F " A ) ) )
526, 51eqtrd 2470 1  |-  ( ( F  Fn  X  /\  A  C_  X )  -> 
( ( x  e.  X ,  y  e.  X  |->  <. ( F `  x ) ,  ( F `  y )
>. ) " ( A  X.  A ) )  =  ( ( F
" A )  X.  ( F " A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   T. wtru 1438    e. wcel 1870   E.wrex 2783   _Vcvv 3087    C_ wss 3442   <.cop 4008    |-> cmpt 4484    X. cxp 4852   ran crn 4855    |` cres 4856   "cima 4857    Fn wfn 5596   ` cfv 5601    |-> cmpt2 6307   1stc1st 6805   2ndc2nd 6806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808
This theorem is referenced by:  fmucnd  21238
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