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Theorem xppreima 27175
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 5616 . . . . 5  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fncnvima2 6002 . . . . 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 465 . . 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 5942 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( 1st  o.  F ) `  x
)  =  ( 1st `  ( F `  x
) ) )
6 fvco 5942 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( 2nd  o.  F ) `  x
)  =  ( 2nd `  ( F `  x
) ) )
75, 6opeq12d 4221 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  <. ( ( 1st  o.  F ) `  x
) ,  ( ( 2nd  o.  F ) `
 x ) >.  =  <. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )
87eqeq2d 2481 . . . . . . . 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 2536 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( 1st 
o.  F ) `  x )  e.  Y  <->  ( 1st `  ( F `
 x ) )  e.  Y ) )
106eleq1d 2536 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( 2nd 
o.  F ) `  x )  e.  Z  <->  ( 2nd `  ( F `
 x ) )  e.  Z ) )
119, 10anbi12d 710 . . . . . . . 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 710 . . . . . . 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 6816 . . . . . . 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 714 . . . . 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 27174 . . . . . 6  |-  ( ( ( Fun  F  /\  ran  F  C_  ( _V  X.  _V ) )  /\  x  e.  dom  F )  ->  ( F `  x )  =  <. ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) >. )
1716biantrurd 508 . . . . 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 6804 . . . . . . . . . . 11  |-  1st : _V -onto-> _V
19 fofun 5795 . . . . . . . . . . 11  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
2018, 19ax-mp 5 . . . . . . . . . 10  |-  Fun  1st
21 funco 5625 . . . . . . . . . 10  |-  ( ( Fun  1st  /\  Fun  F
)  ->  Fun  ( 1st 
o.  F ) )
2220, 21mpan 670 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  ( 1st 
o.  F ) )
2322adantr 465 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  Fun  ( 1st  o.  F
) )
24 ssv 3524 . . . . . . . . . . . 12  |-  ( F
" dom  F )  C_ 
_V
25 fof 5794 . . . . . . . . . . . . 13  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
26 fdm 5734 . . . . . . . . . . . . 13  |-  ( 1st
: _V --> _V  ->  dom 
1st  =  _V )
2718, 25, 26mp2b 10 . . . . . . . . . . . 12  |-  dom  1st  =  _V
2824, 27sseqtr4i 3537 . . . . . . . . . . 11  |-  ( F
" dom  F )  C_ 
dom  1st
29 ssid 3523 . . . . . . . . . . . 12  |-  dom  F  C_ 
dom  F
30 funimass3 5996 . . . . . . . . . . . 12  |-  ( ( Fun  F  /\  dom  F 
C_  dom  F )  ->  ( ( F " dom  F )  C_  dom  1st  <->  dom 
F  C_  ( `' F " dom  1st )
) )
3129, 30mpan2 671 . . . . . . . . . . 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 3504 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  x  e.  ( `' F " dom  1st )
)
34 dmco 5514 . . . . . . . . 9  |-  dom  ( 1st  o.  F )  =  ( `' F " dom  1st )
3533, 34syl6eleqr 2566 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  x  e.  dom  ( 1st 
o.  F ) )
36 fvimacnv 5995 . . . . . . . 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 661 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( 1st 
o.  F ) `  x )  e.  Y  <->  x  e.  ( `' ( 1st  o.  F )
" Y ) ) )
38 fo2nd 6805 . . . . . . . . . . 11  |-  2nd : _V -onto-> _V
39 fofun 5795 . . . . . . . . . . 11  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
4038, 39ax-mp 5 . . . . . . . . . 10  |-  Fun  2nd
41 funco 5625 . . . . . . . . . 10  |-  ( ( Fun  2nd  /\  Fun  F
)  ->  Fun  ( 2nd 
o.  F ) )
4240, 41mpan 670 . . . . . . . . 9  |-  ( Fun 
F  ->  Fun  ( 2nd 
o.  F ) )
4342adantr 465 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  Fun  ( 2nd  o.  F
) )
44 fof 5794 . . . . . . . . . . . . 13  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
45 fdm 5734 . . . . . . . . . . . . 13  |-  ( 2nd
: _V --> _V  ->  dom 
2nd  =  _V )
4638, 44, 45mp2b 10 . . . . . . . . . . . 12  |-  dom  2nd  =  _V
4724, 46sseqtr4i 3537 . . . . . . . . . . 11  |-  ( F
" dom  F )  C_ 
dom  2nd
48 funimass3 5996 . . . . . . . . . . . 12  |-  ( ( Fun  F  /\  dom  F 
C_  dom  F )  ->  ( ( F " dom  F )  C_  dom  2nd  <->  dom 
F  C_  ( `' F " dom  2nd )
) )
4929, 48mpan2 671 . . . . . . . . . . 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 3504 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  x  e.  ( `' F " dom  2nd )
)
52 dmco 5514 . . . . . . . . 9  |-  dom  ( 2nd  o.  F )  =  ( `' F " dom  2nd )
5351, 52syl6eleqr 2566 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  x  e.  dom  ( 2nd 
o.  F ) )
54 fvimacnv 5995 . . . . . . . 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 661 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( 2nd 
o.  F ) `  x )  e.  Z  <->  x  e.  ( `' ( 2nd  o.  F )
" Z ) ) )
5637, 55anbi12d 710 . . . . . 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 714 . . . . 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 3104 . . 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 2508 . 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 3484 . . . 4  |-  ( dom 
F  i^i  ( `' ( 1st  o.  F )
" Y ) )  =  { x  e. 
dom  F  |  x  e.  ( `' ( 1st 
o.  F ) " Y ) }
62 dfin5 3484 . . . 4  |-  ( dom 
F  i^i  ( `' ( 2nd  o.  F )
" Z ) )  =  { x  e. 
dom  F  |  x  e.  ( `' ( 2nd 
o.  F ) " Z ) }
6361, 62ineq12i 3698 . . 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 5356 . . . . . 6  |-  ( `' ( 1st  o.  F
) " Y ) 
C_  dom  ( 1st  o.  F )
65 dmcoss 5261 . . . . . 6  |-  dom  ( 1st  o.  F )  C_  dom  F
6664, 65sstri 3513 . . . . 5  |-  ( `' ( 1st  o.  F
) " Y ) 
C_  dom  F
67 dfss1 3703 . . . . 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 5356 . . . . . 6  |-  ( `' ( 2nd  o.  F
) " Z ) 
C_  dom  ( 2nd  o.  F )
70 dmcoss 5261 . . . . . 6  |-  dom  ( 2nd  o.  F )  C_  dom  F
7169, 70sstri 3513 . . . . 5  |-  ( `' ( 2nd  o.  F
) " Z ) 
C_  dom  F
72 dfss1 3703 . . . . 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 3698 . . 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 3770 . . 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 2505 . 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 2524 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 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    i^i cin 3475    C_ wss 3476   <.cop 4033    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002    o. ccom 5003   Fun wfun 5581    Fn wfn 5582   -->wf 5583   -onto->wfo 5585   ` cfv 5587   1stc1st 6782   2ndc2nd 6783
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 6575
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-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-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 5550  df-fun 5589  df-fn 5590  df-f 5591  df-fo 5593  df-fv 5595  df-1st 6784  df-2nd 6785
This theorem is referenced by:  xppreima2  27176
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