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Theorem xppreima 23461
Description: The preimage of a cross 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 5386 . . . . 5  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fncnvima2 5754 . . . . 5  |-  ( F  Fn  dom  F  -> 
( `' F "
( Y  X.  Z
) )  =  {
x  e.  dom  F  |  ( F `  x )  e.  ( Y  X.  Z ) } )
31, 2sylbi 187 . . . 4  |-  ( Fun 
F  ->  ( `' F " ( Y  X.  Z ) )  =  { x  e.  dom  F  |  ( F `  x )  e.  ( Y  X.  Z ) } )
43adantr 451 . . 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 5702 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( 1st  o.  F ) `  x
)  =  ( 1st `  ( F `  x
) ) )
6 fvco 5702 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( 2nd  o.  F ) `  x
)  =  ( 2nd `  ( F `  x
) ) )
75, 6opeq12d 3906 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  <. ( ( 1st  o.  F ) `  x
) ,  ( ( 2nd  o.  F ) `
 x ) >.  =  <. ( 1st `  ( F `  x )
) ,  ( 2nd `  ( F `  x
) ) >. )
87eqeq2d 2377 . . . . . . . 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 2432 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( 1st 
o.  F ) `  x )  e.  Y  <->  ( 1st `  ( F `
 x ) )  e.  Y ) )
106eleq1d 2432 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( 2nd 
o.  F ) `  x )  e.  Z  <->  ( 2nd `  ( F `
 x ) )  e.  Z ) )
119, 10anbi12d 691 . . . . . . . 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 691 . . . . . . 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 6278 . . . . . . 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 255 . . . . . 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 695 . . . . 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 23460 . . . . . . . 8  |-  ( ( ( Fun  F  /\  ran  F  C_  ( _V  X.  _V ) )  /\  x  e.  dom  F )  ->  ( F `  x )  =  <. ( ( 1st  o.  F
) `  x ) ,  ( ( 2nd 
o.  F ) `  x ) >. )
1716a1d 22 . . . . . . 7  |-  ( ( ( 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 ) >. ) )
1817pm4.71rd 616 . . . . . 6  |-  ( ( ( 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 ) ) ) )
19 fo1st 6266 . . . . . . . . . . . 12  |-  1st : _V -onto-> _V
20 fofun 5558 . . . . . . . . . . . 12  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
2119, 20ax-mp 8 . . . . . . . . . . 11  |-  Fun  1st
22 funco 5395 . . . . . . . . . . 11  |-  ( ( Fun  1st  /\  Fun  F
)  ->  Fun  ( 1st 
o.  F ) )
2321, 22mpan 651 . . . . . . . . . 10  |-  ( Fun 
F  ->  Fun  ( 1st 
o.  F ) )
2423adantr 451 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  Fun  ( 1st  o.  F
) )
25 ssv 3284 . . . . . . . . . . . . 13  |-  ( F
" dom  F )  C_ 
_V
26 fof 5557 . . . . . . . . . . . . . 14  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
27 fdm 5499 . . . . . . . . . . . . . 14  |-  ( 1st
: _V --> _V  ->  dom 
1st  =  _V )
2819, 26, 27mp2b 9 . . . . . . . . . . . . 13  |-  dom  1st  =  _V
2925, 28sseqtr4i 3297 . . . . . . . . . . . 12  |-  ( F
" dom  F )  C_ 
dom  1st
30 ssid 3283 . . . . . . . . . . . . 13  |-  dom  F  C_ 
dom  F
31 funimass3 5748 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\  dom  F 
C_  dom  F )  ->  ( ( F " dom  F )  C_  dom  1st  <->  dom 
F  C_  ( `' F " dom  1st )
) )
3230, 31mpan2 652 . . . . . . . . . . . 12  |-  ( Fun 
F  ->  ( ( F " dom  F ) 
C_  dom  1st  <->  dom  F  C_  ( `' F " dom  1st ) ) )
3329, 32mpbii 202 . . . . . . . . . . 11  |-  ( Fun 
F  ->  dom  F  C_  ( `' F " dom  1st ) )
3433sselda 3266 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  x  e.  ( `' F " dom  1st )
)
35 dmco 5284 . . . . . . . . . 10  |-  dom  ( 1st  o.  F )  =  ( `' F " dom  1st )
3634, 35syl6eleqr 2457 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  x  e.  dom  ( 1st 
o.  F ) )
37 fvimacnv 5747 . . . . . . . . 9  |-  ( ( Fun  ( 1st  o.  F )  /\  x  e.  dom  ( 1st  o.  F ) )  -> 
( ( ( 1st 
o.  F ) `  x )  e.  Y  <->  x  e.  ( `' ( 1st  o.  F )
" Y ) ) )
3824, 36, 37syl2anc 642 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( 1st 
o.  F ) `  x )  e.  Y  <->  x  e.  ( `' ( 1st  o.  F )
" Y ) ) )
39 fo2nd 6267 . . . . . . . . . . . 12  |-  2nd : _V -onto-> _V
40 fofun 5558 . . . . . . . . . . . 12  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
4139, 40ax-mp 8 . . . . . . . . . . 11  |-  Fun  2nd
42 funco 5395 . . . . . . . . . . 11  |-  ( ( Fun  2nd  /\  Fun  F
)  ->  Fun  ( 2nd 
o.  F ) )
4341, 42mpan 651 . . . . . . . . . 10  |-  ( Fun 
F  ->  Fun  ( 2nd 
o.  F ) )
4443adantr 451 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  Fun  ( 2nd  o.  F
) )
45 fof 5557 . . . . . . . . . . . . . 14  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
46 fdm 5499 . . . . . . . . . . . . . 14  |-  ( 2nd
: _V --> _V  ->  dom 
2nd  =  _V )
4739, 45, 46mp2b 9 . . . . . . . . . . . . 13  |-  dom  2nd  =  _V
4825, 47sseqtr4i 3297 . . . . . . . . . . . 12  |-  ( F
" dom  F )  C_ 
dom  2nd
49 funimass3 5748 . . . . . . . . . . . . 13  |-  ( ( Fun  F  /\  dom  F 
C_  dom  F )  ->  ( ( F " dom  F )  C_  dom  2nd  <->  dom 
F  C_  ( `' F " dom  2nd )
) )
5030, 49mpan2 652 . . . . . . . . . . . 12  |-  ( Fun 
F  ->  ( ( F " dom  F ) 
C_  dom  2nd  <->  dom  F  C_  ( `' F " dom  2nd ) ) )
5148, 50mpbii 202 . . . . . . . . . . 11  |-  ( Fun 
F  ->  dom  F  C_  ( `' F " dom  2nd ) )
5251sselda 3266 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  x  e.  ( `' F " dom  2nd )
)
53 dmco 5284 . . . . . . . . . 10  |-  dom  ( 2nd  o.  F )  =  ( `' F " dom  2nd )
5452, 53syl6eleqr 2457 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  x  e.  dom  ( 2nd 
o.  F ) )
55 fvimacnv 5747 . . . . . . . . 9  |-  ( ( Fun  ( 2nd  o.  F )  /\  x  e.  dom  ( 2nd  o.  F ) )  -> 
( ( ( 2nd 
o.  F ) `  x )  e.  Z  <->  x  e.  ( `' ( 2nd  o.  F )
" Z ) ) )
5644, 54, 55syl2anc 642 . . . . . . . 8  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( ( 2nd 
o.  F ) `  x )  e.  Z  <->  x  e.  ( `' ( 2nd  o.  F )
" Z ) ) )
5738, 56anbi12d 691 . . . . . . 7  |-  ( ( 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 ) ) ) )
5857adantlr 695 . . . . . 6  |-  ( ( ( 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 ) ) ) )
5918, 58bitr3d 246 . . . . 5  |-  ( ( ( Fun  F  /\  ran  F  C_  ( _V  X.  _V ) )  /\  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 ) )  <->  ( x  e.  ( `' ( 1st 
o.  F ) " Y )  /\  x  e.  ( `' ( 2nd 
o.  F ) " Z ) ) ) )
6015, 59bitrd 244 . . . 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 ) ) ) )
6160rabbidva 2864 . . 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 ) ) } )
624, 61eqtrd 2398 . 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
) ) } )
63 cnvimass 5136 . . . . . . 7  |-  ( `' ( 1st  o.  F
) " Y ) 
C_  dom  ( 1st  o.  F )
64 dmcoss 5047 . . . . . . 7  |-  dom  ( 1st  o.  F )  C_  dom  F
6563, 64sstri 3274 . . . . . 6  |-  ( `' ( 1st  o.  F
) " Y ) 
C_  dom  F
66 dfss1 3461 . . . . . 6  |-  ( ( `' ( 1st  o.  F ) " Y
)  C_  dom  F  <->  ( dom  F  i^i  ( `' ( 1st  o.  F )
" Y ) )  =  ( `' ( 1st  o.  F )
" Y ) )
6765, 66mpbi 199 . . . . 5  |-  ( dom 
F  i^i  ( `' ( 1st  o.  F )
" Y ) )  =  ( `' ( 1st  o.  F )
" Y )
68 cnvimass 5136 . . . . . . 7  |-  ( `' ( 2nd  o.  F
) " Z ) 
C_  dom  ( 2nd  o.  F )
69 dmcoss 5047 . . . . . . 7  |-  dom  ( 2nd  o.  F )  C_  dom  F
7068, 69sstri 3274 . . . . . 6  |-  ( `' ( 2nd  o.  F
) " Z ) 
C_  dom  F
71 dfss1 3461 . . . . . 6  |-  ( ( `' ( 2nd  o.  F ) " Z
)  C_  dom  F  <->  ( dom  F  i^i  ( `' ( 2nd  o.  F )
" Z ) )  =  ( `' ( 2nd  o.  F )
" Z ) )
7270, 71mpbi 199 . . . . 5  |-  ( dom 
F  i^i  ( `' ( 2nd  o.  F )
" Z ) )  =  ( `' ( 2nd  o.  F )
" Z )
7367, 72ineq12i 3456 . . . 4  |-  ( ( 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
) )
74 dfin5 3246 . . . . 5  |-  ( dom 
F  i^i  ( `' ( 1st  o.  F )
" Y ) )  =  { x  e. 
dom  F  |  x  e.  ( `' ( 1st 
o.  F ) " Y ) }
75 dfin5 3246 . . . . 5  |-  ( dom 
F  i^i  ( `' ( 2nd  o.  F )
" Z ) )  =  { x  e. 
dom  F  |  x  e.  ( `' ( 2nd 
o.  F ) " Z ) }
7674, 75ineq12i 3456 . . . 4  |-  ( ( 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
) } )
7773, 76eqtr3i 2388 . . 3  |-  ( ( `' ( 1st  o.  F ) " Y
)  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 ) } )
78 inrab 3528 . . 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 ) ) }
7977, 78eqtr2i 2387 . 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 ) )
8062, 79syl6eq 2414 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   {crab 2632   _Vcvv 2873    i^i cin 3237    C_ wss 3238   <.cop 3732    X. cxp 4790   `'ccnv 4791   dom cdm 4792   ran crn 4793   "cima 4795    o. ccom 4796   Fun wfun 5352    Fn wfn 5353   -->wf 5354   -onto->wfo 5356   ` cfv 5358   1stc1st 6247   2ndc2nd 6248
This theorem is referenced by:  xppreima2  23462
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fo 5364  df-fv 5366  df-1st 6249  df-2nd 6250
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