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Theorem fmucndlem 19993
Description: Lemma for fmucnd 19994. (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 4956 . . 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 461 . . . . 5  |-  ( ( F  Fn  X  /\  A  C_  X )  ->  A  C_  X )
3 resmpt2 6293 . . . . 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 667 . . . 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 5170 . . 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 2505 . 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 3075 . . . . . . . . . . . . 13  |-  x  e. 
_V
8 vex 3075 . . . . . . . . . . . . 13  |-  y  e. 
_V
97, 8op1std 6692 . . . . . . . . . . . 12  |-  ( p  =  <. x ,  y
>.  ->  ( 1st `  p
)  =  x )
109fveq2d 5798 . . . . . . . . . . 11  |-  ( p  =  <. x ,  y
>.  ->  ( F `  ( 1st `  p ) )  =  ( F `
 x ) )
117, 8op2ndd 6693 . . . . . . . . . . . 12  |-  ( p  =  <. x ,  y
>.  ->  ( 2nd `  p
)  =  y )
1211fveq2d 5798 . . . . . . . . . . 11  |-  ( p  =  <. x ,  y
>.  ->  ( F `  ( 2nd `  p ) )  =  ( F `
 y ) )
1310, 12opeq12d 4170 . . . . . . . . . 10  |-  ( p  =  <. x ,  y
>.  ->  <. ( F `  ( 1st `  p ) ) ,  ( F `
 ( 2nd `  p
) ) >.  =  <. ( F `  x ) ,  ( F `  y ) >. )
1413mpt2mpt 6287 . . . . . . . . 9  |-  ( p  e.  ( A  X.  A )  |->  <. ( F `  ( 1st `  p ) ) ,  ( F `  ( 2nd `  p ) )
>. )  =  (
x  e.  A , 
y  e.  A  |->  <.
( F `  x
) ,  ( F `
 y ) >.
)
1514eqcomi 2465 . . . . . . . 8  |-  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y ) >. )  =  ( p  e.  ( A  X.  A
)  |->  <. ( F `  ( 1st `  p ) ) ,  ( F `
 ( 2nd `  p
) ) >. )
1615rneqi 5169 . . . . . . 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 5804 . . . . . . . 8  |-  ( F `
 ( 1st `  p
) )  e.  _V
1817a1i 11 . . . . . . 7  |-  ( ( T.  /\  p  e.  ( A  X.  A
) )  ->  ( F `  ( 1st `  p ) )  e. 
_V )
19 fvex 5804 . . . . . . . 8  |-  ( F `
 ( 2nd `  p
) )  e.  _V
2019a1i 11 . . . . . . 7  |-  ( ( T.  /\  p  e.  ( A  X.  A
) )  ->  ( F `  ( 2nd `  p ) )  e. 
_V )
2116, 18, 20fliftrel 6105 . . . . . 6  |-  ( T. 
->  ran  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y )
>. )  C_  ( _V 
X.  _V ) )
2221trud 1379 . . . . 5  |-  ran  (
x  e.  A , 
y  e.  A  |->  <.
( F `  x
) ,  ( F `
 y ) >.
)  C_  ( _V  X.  _V )
2322sseli 3455 . . . 4  |-  ( p  e.  ran  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y ) >. )  ->  p  e.  ( _V 
X.  _V ) )
2423adantl 466 . . 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 5049 . . . . 5  |-  ( ( F " A )  X.  ( F " A ) )  C_  ( _V  X.  _V )
2625sseli 3455 . . . 4  |-  ( p  e.  ( ( F
" A )  X.  ( F " A
) )  ->  p  e.  ( _V  X.  _V ) )
2726adantl 466 . . 3  |-  ( ( ( F  Fn  X  /\  A  C_  X )  /\  p  e.  ( ( F " A
)  X.  ( F
" A ) ) )  ->  p  e.  ( _V  X.  _V )
)
28 fvelimab 5851 . . . . . . . 8  |-  ( ( F  Fn  X  /\  A  C_  X )  -> 
( ( 1st `  p
)  e.  ( F
" A )  <->  E. x  e.  A  ( F `  x )  =  ( 1st `  p ) ) )
29 fvelimab 5851 . . . . . . . 8  |-  ( ( F  Fn  X  /\  A  C_  X )  -> 
( ( 2nd `  p
)  e.  ( F
" A )  <->  E. y  e.  A  ( F `  y )  =  ( 2nd `  p ) ) )
3028, 29anbi12d 710 . . . . . . 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 2452 . . . . . . . . 9  |-  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y ) >. )  =  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y )
>. )
32 opex 4659 . . . . . . . . 9  |-  <. ( F `  x ) ,  ( F `  y ) >.  e.  _V
3331, 32elrnmpt2 6308 . . . . . . . 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 2461 . . . . . . . . . 10  |-  ( <.
( 1st `  p
) ,  ( 2nd `  p ) >.  =  <. ( F `  x ) ,  ( F `  y ) >.  <->  <. ( F `
 x ) ,  ( F `  y
) >.  =  <. ( 1st `  p ) ,  ( 2nd `  p
) >. )
35 fvex 5804 . . . . . . . . . . 11  |-  ( 1st `  p )  e.  _V
36 fvex 5804 . . . . . . . . . . 11  |-  ( 2nd `  p )  e.  _V
3735, 36opth2 4673 . . . . . . . . . 10  |-  ( <.
( F `  x
) ,  ( F `
 y ) >.  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  <->  ( ( F `  x )  =  ( 1st `  p
)  /\  ( F `  y )  =  ( 2nd `  p ) ) )
3834, 37bitri 249 . . . . . . . . 9  |-  ( <.
( 1st `  p
) ,  ( 2nd `  p ) >.  =  <. ( F `  x ) ,  ( F `  y ) >.  <->  ( ( F `  x )  =  ( 1st `  p
)  /\  ( F `  y )  =  ( 2nd `  p ) ) )
39382rexbii 2863 . . . . . . . 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 2988 . . . . . . . 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 271 . . . . . . 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 264 . . . . . 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 4972 . . . . . 6  |-  ( <.
( 1st `  p
) ,  ( 2nd `  p ) >.  e.  ( ( F " A
)  X.  ( F
" A ) )  <-> 
( ( 1st `  p
)  e.  ( F
" A )  /\  ( 2nd `  p )  e.  ( F " A ) ) )
4442, 43syl6bbr 263 . . . . 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 465 . . . 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 6718 . . . . . 6  |-  ( p  e.  ( _V  X.  _V )  ->  p  = 
<. ( 1st `  p
) ,  ( 2nd `  p ) >. )
4746adantl 466 . . . . 5  |-  ( ( ( F  Fn  X  /\  A  C_  X )  /\  p  e.  ( _V  X.  _V )
)  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
4847eleq1d 2521 . . . 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 2521 . . . 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 285 . . 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 2450 . 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 2493 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 184    /\ wa 369    = wceq 1370   T. wtru 1371    e. wcel 1758   E.wrex 2797   _Vcvv 3072    C_ wss 3431   <.cop 3986    |-> cmpt 4453    X. cxp 4941   ran crn 4944    |` cres 4945   "cima 4946    Fn wfn 5516   ` cfv 5521    |-> cmpt2 6197   1stc1st 6680   2ndc2nd 6681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fv 5529  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683
This theorem is referenced by:  fmucnd  19994
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