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Theorem xppreima 25899
Description: The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 6-Jun-2017.)
Assertion
Ref Expression
xppreima  |-  ( ( Fun  F  /\  ran  F 
C_  ( _V  X.  _V ) )  ->  ( `' F " ( Y  X.  Z ) )  =  ( ( `' ( 1st  o.  F
) " Y )  i^i  ( `' ( 2nd  o.  F )
" Z ) ) )

Proof of Theorem xppreima
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 funfn 5444 . . . . 5  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fncnvima2 5822 . . . . 5  |-  ( F  Fn  dom  F  -> 
( `' F "
( Y  X.  Z
) )  =  {
x  e.  dom  F  |  ( F `  x )  e.  ( Y  X.  Z ) } )
31, 2sylbi 195 . . . 4  |-  ( Fun 
F  ->  ( `' F " ( Y  X.  Z ) )  =  { x  e.  dom  F  |  ( F `  x )  e.  ( Y  X.  Z ) } )
43adantr 462 . . 3  |-  ( ( Fun  F  /\  ran  F 
C_  ( _V  X.  _V ) )  ->  ( `' F " ( Y  X.  Z ) )  =  { x  e. 
dom  F  |  ( F `  x )  e.  ( Y  X.  Z
) } )
5 fvco 5764 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( 1st  o.  F ) `  x
)  =  ( 1st `  ( F `  x
) ) )
6 fvco 5764 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( 2nd  o.  F ) `  x
)  =  ( 2nd `  ( F `  x
) ) )
75, 6opeq12d 4064 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  <. ( ( 1st  o.  F ) `  x
) ,  ( ( 2nd  o.  F ) `
 x ) >.  =  <. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )
87eqeq2d 2452 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  =  <. ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) >.  <->  ( F `  x )  =  <. ( 1st `  ( F `
 x ) ) ,  ( 2nd `  ( F `  x )
) >. ) )
95eleq1d 2507 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( 1st 
o.  F ) `  x )  e.  Y  <->  ( 1st `  ( F `
 x ) )  e.  Y ) )
106eleq1d 2507 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( 2nd 
o.  F ) `  x )  e.  Z  <->  ( 2nd `  ( F `
 x ) )  e.  Z ) )
119, 10anbi12d 705 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( ( 1st  o.  F ) `
 x )  e.  Y  /\  ( ( 2nd  o.  F ) `
 x )  e.  Z )  <->  ( ( 1st `  ( F `  x ) )  e.  Y  /\  ( 2nd `  ( F `  x
) )  e.  Z
) ) )
128, 11anbi12d 705 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( F `
 x )  = 
<. ( ( 1st  o.  F ) `  x
) ,  ( ( 2nd  o.  F ) `
 x ) >.  /\  ( ( ( 1st 
o.  F ) `  x )  e.  Y  /\  ( ( 2nd  o.  F ) `  x
)  e.  Z ) )  <->  ( ( F `
 x )  = 
<. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >.  /\  (
( 1st `  ( F `  x )
)  e.  Y  /\  ( 2nd `  ( F `
 x ) )  e.  Z ) ) ) )
13 elxp6 6607 . . . . . . 7  |-  ( ( F `  x )  e.  ( Y  X.  Z )  <->  ( ( F `  x )  =  <. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >.  /\  (
( 1st `  ( F `  x )
)  e.  Y  /\  ( 2nd `  ( F `
 x ) )  e.  Z ) ) )
1412, 13syl6rbbr 264 . . . . . 6  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  e.  ( Y  X.  Z )  <-> 
( ( F `  x )  =  <. ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) >.  /\  (
( ( 1st  o.  F ) `  x
)  e.  Y  /\  ( ( 2nd  o.  F ) `  x
)  e.  Z ) ) ) )
1514adantlr 709 . . . . 5  |-  ( ( ( Fun  F  /\  ran  F  C_  ( _V  X.  _V ) )  /\  x  e.  dom  F )  ->  ( ( F `
 x )  e.  ( Y  X.  Z
)  <->  ( ( F `
 x )  = 
<. ( ( 1st  o.  F ) `  x
) ,  ( ( 2nd  o.  F ) `
 x ) >.  /\  ( ( ( 1st 
o.  F ) `  x )  e.  Y  /\  ( ( 2nd  o.  F ) `  x
)  e.  Z ) ) ) )
16 opfv 25898 . . . . . 6  |-  ( ( ( Fun  F  /\  ran  F  C_  ( _V  X.  _V ) )  /\  x  e.  dom  F )  ->  ( F `  x )  =  <. ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) >. )
1716biantrurd 505 . . . . 5  |-  ( ( ( Fun  F  /\  ran  F  C_  ( _V  X.  _V ) )  /\  x  e.  dom  F )  ->  ( ( ( ( 1st  o.  F
) `  x )  e.  Y  /\  (
( 2nd  o.  F
) `  x )  e.  Z )  <->  ( ( F `  x )  =  <. ( ( 1st 
o.  F ) `  x ) ,  ( ( 2nd  o.  F
) `  x ) >.  /\  ( ( ( 1st  o.  F ) `
 x )  e.  Y  /\  ( ( 2nd  o.  F ) `
 x )  e.  Z ) ) ) )
18 fo1st 6595 . . . . . . . . . . 11  |-  1st : _V -onto-> _V
19 fofun 5618 . . . . . . . . . . 11  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
2018, 19ax-mp 5 . . . . . . . . . 10  |-  Fun  1st
21 funco 5453 . . . . . . . . . 10  |-  ( ( Fun  1st  /\  Fun  F
)  ->  Fun  ( 1st 
o.  F ) )
2220, 21mpan 665 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  ( 1st 
o.  F ) )
2322adantr 462 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  Fun  ( 1st  o.  F
) )
24 ssv 3373 . . . . . . . . . . . 12  |-  ( F
" dom  F )  C_ 
_V
25 fof 5617 . . . . . . . . . . . . 13  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
26 fdm 5560 . . . . . . . . . . . . 13  |-  ( 1st
: _V --> _V  ->  dom 
1st  =  _V )
2718, 25, 26mp2b 10 . . . . . . . . . . . 12  |-  dom  1st  =  _V
2824, 27sseqtr4i 3386 . . . . . . . . . . 11  |-  ( F
" dom  F )  C_ 
dom  1st
29 ssid 3372 . . . . . . . . . . . 12  |-  dom  F  C_ 
dom  F
30 funimass3 5816 . . . . . . . . . . . 12  |-  ( ( Fun  F  /\  dom  F 
C_  dom  F )  ->  ( ( F " dom  F )  C_  dom  1st  <->  dom 
F  C_  ( `' F " dom  1st )
) )
3129, 30mpan2 666 . . . . . . . . . . 11  |-  ( Fun 
F  ->  ( ( F " dom  F ) 
C_  dom  1st  <->  dom  F  C_  ( `' F " dom  1st ) ) )
3228, 31mpbii 211 . . . . . . . . . 10  |-  ( Fun 
F  ->  dom  F  C_  ( `' F " dom  1st ) )
3332sselda 3353 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  x  e.  ( `' F " dom  1st )
)
34 dmco 5343 . . . . . . . . 9  |-  dom  ( 1st  o.  F )  =  ( `' F " dom  1st )
3533, 34syl6eleqr 2532 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  x  e.  dom  ( 1st 
o.  F ) )
36 fvimacnv 5815 . . . . . . . 8  |-  ( ( Fun  ( 1st  o.  F )  /\  x  e.  dom  ( 1st  o.  F ) )  -> 
( ( ( 1st 
o.  F ) `  x )  e.  Y  <->  x  e.  ( `' ( 1st  o.  F )
" Y ) ) )
3723, 35, 36syl2anc 656 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( 1st 
o.  F ) `  x )  e.  Y  <->  x  e.  ( `' ( 1st  o.  F )
" Y ) ) )
38 fo2nd 6596 . . . . . . . . . . 11  |-  2nd : _V -onto-> _V
39 fofun 5618 . . . . . . . . . . 11  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
4038, 39ax-mp 5 . . . . . . . . . 10  |-  Fun  2nd
41 funco 5453 . . . . . . . . . 10  |-  ( ( Fun  2nd  /\  Fun  F
)  ->  Fun  ( 2nd 
o.  F ) )
4240, 41mpan 665 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  ( 2nd 
o.  F ) )
4342adantr 462 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  Fun  ( 2nd  o.  F
) )
44 fof 5617 . . . . . . . . . . . . 13  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
45 fdm 5560 . . . . . . . . . . . . 13  |-  ( 2nd
: _V --> _V  ->  dom 
2nd  =  _V )
4638, 44, 45mp2b 10 . . . . . . . . . . . 12  |-  dom  2nd  =  _V
4724, 46sseqtr4i 3386 . . . . . . . . . . 11  |-  ( F
" dom  F )  C_ 
dom  2nd
48 funimass3 5816 . . . . . . . . . . . 12  |-  ( ( Fun  F  /\  dom  F 
C_  dom  F )  ->  ( ( F " dom  F )  C_  dom  2nd  <->  dom 
F  C_  ( `' F " dom  2nd )
) )
4929, 48mpan2 666 . . . . . . . . . . 11  |-  ( Fun 
F  ->  ( ( F " dom  F ) 
C_  dom  2nd  <->  dom  F  C_  ( `' F " dom  2nd ) ) )
5047, 49mpbii 211 . . . . . . . . . 10  |-  ( Fun 
F  ->  dom  F  C_  ( `' F " dom  2nd ) )
5150sselda 3353 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  x  e.  ( `' F " dom  2nd )
)
52 dmco 5343 . . . . . . . . 9  |-  dom  ( 2nd  o.  F )  =  ( `' F " dom  2nd )
5351, 52syl6eleqr 2532 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  x  e.  dom  ( 2nd 
o.  F ) )
54 fvimacnv 5815 . . . . . . . 8  |-  ( ( Fun  ( 2nd  o.  F )  /\  x  e.  dom  ( 2nd  o.  F ) )  -> 
( ( ( 2nd 
o.  F ) `  x )  e.  Z  <->  x  e.  ( `' ( 2nd  o.  F )
" Z ) ) )
5543, 53, 54syl2anc 656 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( 2nd 
o.  F ) `  x )  e.  Z  <->  x  e.  ( `' ( 2nd  o.  F )
" Z ) ) )
5637, 55anbi12d 705 . . . . . 6  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( ( 1st  o.  F ) `
 x )  e.  Y  /\  ( ( 2nd  o.  F ) `
 x )  e.  Z )  <->  ( x  e.  ( `' ( 1st 
o.  F ) " Y )  /\  x  e.  ( `' ( 2nd 
o.  F ) " Z ) ) ) )
5756adantlr 709 . . . . 5  |-  ( ( ( Fun  F  /\  ran  F  C_  ( _V  X.  _V ) )  /\  x  e.  dom  F )  ->  ( ( ( ( 1st  o.  F
) `  x )  e.  Y  /\  (
( 2nd  o.  F
) `  x )  e.  Z )  <->  ( x  e.  ( `' ( 1st 
o.  F ) " Y )  /\  x  e.  ( `' ( 2nd 
o.  F ) " Z ) ) ) )
5815, 17, 573bitr2d 281 . . . 4  |-  ( ( ( Fun  F  /\  ran  F  C_  ( _V  X.  _V ) )  /\  x  e.  dom  F )  ->  ( ( F `
 x )  e.  ( Y  X.  Z
)  <->  ( x  e.  ( `' ( 1st 
o.  F ) " Y )  /\  x  e.  ( `' ( 2nd 
o.  F ) " Z ) ) ) )
5958rabbidva 2961 . . 3  |-  ( ( Fun  F  /\  ran  F 
C_  ( _V  X.  _V ) )  ->  { x  e.  dom  F  |  ( F `  x )  e.  ( Y  X.  Z ) }  =  { x  e.  dom  F  |  ( x  e.  ( `' ( 1st 
o.  F ) " Y )  /\  x  e.  ( `' ( 2nd 
o.  F ) " Z ) ) } )
604, 59eqtrd 2473 . 2  |-  ( ( Fun  F  /\  ran  F 
C_  ( _V  X.  _V ) )  ->  ( `' F " ( Y  X.  Z ) )  =  { x  e. 
dom  F  |  (
x  e.  ( `' ( 1st  o.  F
) " Y )  /\  x  e.  ( `' ( 2nd  o.  F ) " Z
) ) } )
61 dfin5 3333 . . . 4  |-  ( dom 
F  i^i  ( `' ( 1st  o.  F )
" Y ) )  =  { x  e. 
dom  F  |  x  e.  ( `' ( 1st 
o.  F ) " Y ) }
62 dfin5 3333 . . . 4  |-  ( dom 
F  i^i  ( `' ( 2nd  o.  F )
" Z ) )  =  { x  e. 
dom  F  |  x  e.  ( `' ( 2nd 
o.  F ) " Z ) }
6361, 62ineq12i 3547 . . 3  |-  ( ( dom  F  i^i  ( `' ( 1st  o.  F ) " Y
) )  i^i  ( dom  F  i^i  ( `' ( 2nd  o.  F
) " Z ) ) )  =  ( { x  e.  dom  F  |  x  e.  ( `' ( 1st  o.  F ) " Y
) }  i^i  {
x  e.  dom  F  |  x  e.  ( `' ( 2nd  o.  F ) " Z
) } )
64 cnvimass 5186 . . . . . 6  |-  ( `' ( 1st  o.  F
) " Y ) 
C_  dom  ( 1st  o.  F )
65 dmcoss 5095 . . . . . 6  |-  dom  ( 1st  o.  F )  C_  dom  F
6664, 65sstri 3362 . . . . 5  |-  ( `' ( 1st  o.  F
) " Y ) 
C_  dom  F
67 dfss1 3552 . . . . 5  |-  ( ( `' ( 1st  o.  F ) " Y
)  C_  dom  F  <->  ( dom  F  i^i  ( `' ( 1st  o.  F )
" Y ) )  =  ( `' ( 1st  o.  F )
" Y ) )
6866, 67mpbi 208 . . . 4  |-  ( dom 
F  i^i  ( `' ( 1st  o.  F )
" Y ) )  =  ( `' ( 1st  o.  F )
" Y )
69 cnvimass 5186 . . . . . 6  |-  ( `' ( 2nd  o.  F
) " Z ) 
C_  dom  ( 2nd  o.  F )
70 dmcoss 5095 . . . . . 6  |-  dom  ( 2nd  o.  F )  C_  dom  F
7169, 70sstri 3362 . . . . 5  |-  ( `' ( 2nd  o.  F
) " Z ) 
C_  dom  F
72 dfss1 3552 . . . . 5  |-  ( ( `' ( 2nd  o.  F ) " Z
)  C_  dom  F  <->  ( dom  F  i^i  ( `' ( 2nd  o.  F )
" Z ) )  =  ( `' ( 2nd  o.  F )
" Z ) )
7371, 72mpbi 208 . . . 4  |-  ( dom 
F  i^i  ( `' ( 2nd  o.  F )
" Z ) )  =  ( `' ( 2nd  o.  F )
" Z )
7468, 73ineq12i 3547 . . 3  |-  ( ( dom  F  i^i  ( `' ( 1st  o.  F ) " Y
) )  i^i  ( dom  F  i^i  ( `' ( 2nd  o.  F
) " Z ) ) )  =  ( ( `' ( 1st 
o.  F ) " Y )  i^i  ( `' ( 2nd  o.  F ) " Z
) )
75 inrab 3619 . . 3  |-  ( { x  e.  dom  F  |  x  e.  ( `' ( 1st  o.  F ) " Y
) }  i^i  {
x  e.  dom  F  |  x  e.  ( `' ( 2nd  o.  F ) " Z
) } )  =  { x  e.  dom  F  |  ( x  e.  ( `' ( 1st 
o.  F ) " Y )  /\  x  e.  ( `' ( 2nd 
o.  F ) " Z ) ) }
7663, 74, 753eqtr3ri 2470 . 2  |-  { x  e.  dom  F  |  ( x  e.  ( `' ( 1st  o.  F
) " Y )  /\  x  e.  ( `' ( 2nd  o.  F ) " Z
) ) }  =  ( ( `' ( 1st  o.  F )
" Y )  i^i  ( `' ( 2nd 
o.  F ) " Z ) )
7760, 76syl6eq 2489 1  |-  ( ( Fun  F  /\  ran  F 
C_  ( _V  X.  _V ) )  ->  ( `' F " ( Y  X.  Z ) )  =  ( ( `' ( 1st  o.  F
) " Y )  i^i  ( `' ( 2nd  o.  F )
" Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   {crab 2717   _Vcvv 2970    i^i cin 3324    C_ wss 3325   <.cop 3880    X. cxp 4834   `'ccnv 4835   dom cdm 4836   ran crn 4837   "cima 4839    o. ccom 4840   Fun wfun 5409    Fn wfn 5410   -->wf 5411   -onto->wfo 5413   ` cfv 5415   1stc1st 6574   2ndc2nd 6575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fo 5421  df-fv 5423  df-1st 6576  df-2nd 6577
This theorem is referenced by:  xppreima2  25900
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