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Theorem fmucndlem 20529
Description: Lemma for fmucnd 20530. (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 5012 . . 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 6382 . . . . 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 5228 . . 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 2520 . 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 3116 . . . . . . . . . . . . 13  |-  x  e. 
_V
8 vex 3116 . . . . . . . . . . . . 13  |-  y  e. 
_V
97, 8op1std 6791 . . . . . . . . . . . 12  |-  ( p  =  <. x ,  y
>.  ->  ( 1st `  p
)  =  x )
109fveq2d 5868 . . . . . . . . . . 11  |-  ( p  =  <. x ,  y
>.  ->  ( F `  ( 1st `  p ) )  =  ( F `
 x ) )
117, 8op2ndd 6792 . . . . . . . . . . . 12  |-  ( p  =  <. x ,  y
>.  ->  ( 2nd `  p
)  =  y )
1211fveq2d 5868 . . . . . . . . . . 11  |-  ( p  =  <. x ,  y
>.  ->  ( F `  ( 2nd `  p ) )  =  ( F `
 y ) )
1310, 12opeq12d 4221 . . . . . . . . . 10  |-  ( p  =  <. x ,  y
>.  ->  <. ( F `  ( 1st `  p ) ) ,  ( F `
 ( 2nd `  p
) ) >.  =  <. ( F `  x ) ,  ( F `  y ) >. )
1413mpt2mpt 6376 . . . . . . . . 9  |-  ( p  e.  ( A  X.  A )  |->  <. ( F `  ( 1st `  p ) ) ,  ( F `  ( 2nd `  p ) )
>. )  =  (
x  e.  A , 
y  e.  A  |->  <.
( F `  x
) ,  ( F `
 y ) >.
)
1514eqcomi 2480 . . . . . . . 8  |-  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y ) >. )  =  ( p  e.  ( A  X.  A
)  |->  <. ( F `  ( 1st `  p ) ) ,  ( F `
 ( 2nd `  p
) ) >. )
1615rneqi 5227 . . . . . . 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 5874 . . . . . . . 8  |-  ( F `
 ( 1st `  p
) )  e.  _V
1817a1i 11 . . . . . . 7  |-  ( ( T.  /\  p  e.  ( A  X.  A
) )  ->  ( F `  ( 1st `  p ) )  e. 
_V )
19 fvex 5874 . . . . . . . 8  |-  ( F `
 ( 2nd `  p
) )  e.  _V
2019a1i 11 . . . . . . 7  |-  ( ( T.  /\  p  e.  ( A  X.  A
) )  ->  ( F `  ( 2nd `  p ) )  e. 
_V )
2116, 18, 20fliftrel 6192 . . . . . 6  |-  ( T. 
->  ran  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y )
>. )  C_  ( _V 
X.  _V ) )
2221trud 1388 . . . . 5  |-  ran  (
x  e.  A , 
y  e.  A  |->  <.
( F `  x
) ,  ( F `
 y ) >.
)  C_  ( _V  X.  _V )
2322sseli 3500 . . . 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 5107 . . . . 5  |-  ( ( F " A )  X.  ( F " A ) )  C_  ( _V  X.  _V )
2625sseli 3500 . . . 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 5921 . . . . . . . 8  |-  ( ( F  Fn  X  /\  A  C_  X )  -> 
( ( 1st `  p
)  e.  ( F
" A )  <->  E. x  e.  A  ( F `  x )  =  ( 1st `  p ) ) )
29 fvelimab 5921 . . . . . . . 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 2467 . . . . . . . . 9  |-  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y ) >. )  =  ( x  e.  A ,  y  e.  A  |->  <. ( F `  x ) ,  ( F `  y )
>. )
32 opex 4711 . . . . . . . . 9  |-  <. ( F `  x ) ,  ( F `  y ) >.  e.  _V
3331, 32elrnmpt2 6397 . . . . . . . 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 2476 . . . . . . . . . 10  |-  ( <.
( 1st `  p
) ,  ( 2nd `  p ) >.  =  <. ( F `  x ) ,  ( F `  y ) >.  <->  <. ( F `
 x ) ,  ( F `  y
) >.  =  <. ( 1st `  p ) ,  ( 2nd `  p
) >. )
35 fvex 5874 . . . . . . . . . . 11  |-  ( 1st `  p )  e.  _V
36 fvex 5874 . . . . . . . . . . 11  |-  ( 2nd `  p )  e.  _V
3735, 36opth2 4725 . . . . . . . . . 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 2966 . . . . . . . 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 3029 . . . . . . . 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 5028 . . . . . 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 6818 . . . . . 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 2536 . . . 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 2536 . . . 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 2465 . 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 2508 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 1379   T. wtru 1380    e. wcel 1767   E.wrex 2815   _Vcvv 3113    C_ wss 3476   <.cop 4033    |-> cmpt 4505    X. cxp 4997   ran crn 5000    |` cres 5001   "cima 5002    Fn wfn 5581   ` cfv 5586    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782
This theorem is referenced by:  fmucnd  20530
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